Hyponormal quantization of planar domains : exponential transform in dimension two

Lecture Notes in Mathematics 2199

Bjoumlrn GustafssonMihai Putinar

Hyponormal Quantization of Planar DomainsExponential Transform in Dimension Two

Editors-in-ChiefJean-Michel Morel CachanBernard Teissier Paris

Advisory BoardMichel Brion GrenobleCamillo De Lellis ZurichAlessio Figalli ZurichDavar Khoshnevisan Salt Lake CityIoannis Kontoyiannis AthensGaacutebor Lugosi BarcelonaMark Podolskij AarhusSylvia Serfaty New YorkAnna Wienhard Heidelberg

More information about this series at httpwwwspringercomseries304

BjRorn Gustafsson bull Mihai Putinar

Hyponormal Quantizationof Planar DomainsExponential Transform in Dimension Two

BjRorn GustafssonDepartment of MathematicsKTH Royal Institute of TechnologyStockholm Sweden

Mihai PutinarMathematics DepartmentUniversity of CaliforniaSanta Barbara CA USA

School of Mathematics Statisticsand Physics

Newcastle UniversityUK

ISSN 0075-8434 ISSN 1617-9692 (electronic)Lecture Notes in MathematicsISBN 978-3-319-65809-4 ISBN 978-3-319-65810-0 (eBook)DOI 101007978-3-319-65810-0

Library of Congress Control Number 2017952198

Mathematics Subject Classification (2010) Primary 47B20 Secondary 30C15 31A25 35Q15 44A60

copy Springer International Publishing AG 2017This work is subject to copyright All rights are reserved by the Publisher whether the whole or part ofthe material is concerned specifically the rights of translation reprinting reuse of illustrations recitationbroadcasting reproduction on microfilms or in any other physical way and transmission or informationstorage and retrieval electronic adaptation computer software or by similar or dissimilar methodologynow known or hereafter developedThe use of general descriptive names registered names trademarks service marks etc in this publicationdoes not imply even in the absence of a specific statement that such names are exempt from the relevantprotective laws and regulations and therefore free for general useThe publisher the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication Neither the publisher nor the authors orthe editors give a warranty express or implied with respect to the material contained herein or for anyerrors or omissions that may have been made The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is Gewerbestrasse 11 6330 Cham Switzerland

Dedicated to Harold S Shapiro on theoccasion of his 90th birthday

Preface

A physicist a mathematician and a computer engineer look at a simple drawingThe physicist immediately sees it as the shape of a melting material with fjordsin formation with instability and bifurcations in the dynamics of the shrinkingshape The mathematician recognizes a real algebraic curve lying on a Riemannsurface belonging to a family of deformations with singularities cusps and causticsin its fibres The engineer happily decodes the picture in the frequency domainidentifying encrypted messages ready to be processed simplified and tuned Thepresent lecture notes touch such topics remaining however on the mathematicalside of the mirror

During the last two decades the authors of this essay have contemplated variousaspects of quadrature domains a class of basic semi-algebraic sets in the complexplane As more and more intricate correspondences between potential theory fluidmechanics and operator theory were unveiled on this ground a need of unifyingvarious approximation schemes emerged This is the broad mathematical theme ofthese notes

Having the advanced understanding of the asymptotics of complex orthogonalpolynomials or of finite central truncations of Toeplitz matrices as a solid basisof comparison we propose a novel approximation scheme for two-dimensional(shaded) domains bearing similarities and dissonances to the classical results Ournumerical experiments point out to yet another skeleton different than already stud-ied potential theoretic bodies The quantization of a shaded domain by a hyponormaloperator with rank-one self-commutator is opening a new perspective to Hilbertspace methods in the analysis of planar shapes Performing the approximation onan exotic space which is not a Lebesgue space associated with a canonical positivemeasure is both intriguing and challenging We stress that in these notes we do nottreat classical orthogonal polynomials in the complex domain such as those derivedfrom weighted Bergman or Hardy spaces

A warning to the reader these are merely lectures notes demarcating a new andmostly uncharted territory Our aim is to open a new vista on the mathematicalaspects (analysis geometry approximation encoding) of 2D shapes carrying adegree of grey function The notes contain a good proportion of original results

viii Preface

or at least a novel arrangement of prior facts proved over two decades by the twoauthors Of course much remains to be done on both theoretical and numerical sidessome pertinent open questions are raised throughout the text

During the rather long gestation of these notes we have benefited from thecontinuous support of our friend and old-time collaborator Nikos Stylianopoulos ofthe University of Cyprus at Nicosia He has produced with unmatched professionalskill the numerical experiments and many illustrations embedded into the text Weare grateful to him for his unconditional assistance We also thank the anonymousreferees for their criticism and encouragement

Our collaborative work could not have been completed without the generous sup-port of several institutions The Royal Institute of Technology of Sweden Universityof California at Santa Barbara University of Cyprus Newcastle University (UK)and Nanyang Technological University of Singapore

Stockholm Sweden Bjoumlrn GustafssonSanta Barbara CA USA Mihai PutinarJune 22 2017

Contents

1 Introduction 1

2 The Exponential Transform 721 Basic Definitions 722 Moments 1123 Positive Definiteness Properties 1324 The Exponential Transform as a Section of a Line Bundle 1625 A Riemann-Hilbert Problem 18

3 Hilbert Space Factorization 2331 Definitions and Generalities 2332 Restrictions and Extensions 2633 Linear Operators on H ˝ 2734 A Functional Model for Hyponormal Operators 3035 Summary in Abstract Setting 3136 The Analytic Subspace Ha˝ 3237 The Analytic Model 3438 A Formal Comparison to Quantum Field Theory 3539 Silva-Koumlthe-Grothendieck Duality 37310 Quadrature Domains 40311 Analytic Functionals 43

4 Exponential Orthogonal Polynomials 4741 Orthogonal Expansions 4742 Zeros of Orthogonal Polynomials 5043 The Hessenberg Matrices 5244 The Matrix Model of Quadrature Domains 54

5 Finite Central Truncations of Linear Operators 5751 Trace Class Perturbations 5752 Padeacute Approximation Scheme 6053 Three Term Relation for the Orthogonal Polynomials 6354 Disjoint Unions of Domains 66

x Contents

55 Perturbations of Finite Truncations 6856 Real Central Truncations 74

6 Mother Bodies 7761 General 7762 Some General Properties of Mother Bodies 8163 Reduction of Inner Product to Mother Body 8464 Regularity of Some Free Boundaries 8665 Procedures for Finding Mother Bodies 89

7 Examples 9371 The Unit Disk 9372 The Annulus 9473 Complements of Unbounded Quadrature Domains 95

731 The Ellipse 97732 The Hypocycloid 99

74 Lemniscates 10375 Polygons 105

751 Computation of Mother Body 105752 Numerical Experiments 105

76 The Half-Disk and Disk with a Sector Removed 108761 Computation of Mother Body 108762 Numerical Experiment 110

77 Domain Bounded by Two Circular Arcs 111771 Numerical Experiment 112

78 External Disk 112781 Numerical Experiment Ellipse Plus Disk 113782 Numerical Experiment Pentagon Plus Disk 113

79 Abelian Domains 115710 Disjoint Union of a Hexagon and a Hypocycloid 116

7101 Numerical Experiment 116711 A Square with a Disk Removed 117

7111 Numerical Experiment 117

8 Comparison with Classical Function Spaces 11981 Bergman Space 11982 Faber Polynomials 120

A Hyponormal Operators 125

Historical Notes 135

Glossary 139

References 141

Index 147

Chapter 1Introduction

Abstract This chapter offers an overview of the main interlacing themes withemphasis on historical development and identification of the original sources

When looking at a picture through a mathematical lens several intrinsic innerskeletons pop up They are sometimes called in a suggestive and colorful way theldquomother bodyrdquo or ldquomadonna bodyrdquo or ldquothe ridgerdquo or ldquocausticrdquo of the originalpicture Geometric or analytic features characterize these skeletons but in generalthey remain very shy displaying their shapes and qualitative features only afterchallenging technical obstacles are resolved For example the natural skeleton of adisk is its center the interval between the foci of an ellipse stands out as a canonicalridge For reasons to be discussed in these notes the internal bisector segments in atriangle form its ldquomother bodyrdquo Disagreement starts with a sector of a disk wherethe ldquobodiesrdquo (maternal madonna type or the one proposed below) are different Inthe scenario outlined in these notes a disjoint union of disks has their centers as anatural skeleton

We start with the relevant question for nowadays visual civilization how totreat mathematically a two dimensional colored picture There are various ways ofencoding an image carrying various degrees of color A common approach havingthe geometric tomographic data as a source is to arrange them into a string ofcomplex numbers called moments

Mk` DZC

zkz`gzdAz 0 k ` lt N

Above z 2 C is the complex variable dA is Lebesgue measure in C g 2 L1Cis a function of compact support and range into Œ0 1 (the shade function) and Ncan be infinity The ideal scenario of possessing complete information (ie N D1) permits a full reconstruction of the shaded image usually relying on stablealgorithms derived from integral operators inversion (Radon Fourier or Laplacetransforms) We do not touch in the present text such reconstruction methods

Assume the more realistic scenario that we know only finitely many momentsthat is N lt 1 The first observation going back to the early works of Mark Kreinis that working in a prescribed window that is replacing above dA by the Lebesguemeasure supported by a compact set K C one can exactly recover the shade func-

copy Springer International Publishing AG 2017B Gustafsson M Putinar Hyponormal Quantization of Planar DomainsLecture Notes in Mathematics 2199 DOI 101007978-3-319-65810-0_1

2 1 Introduction

tion g from moments Mk`N1k`D0 if and only if g has only values 0 and 1 and is the

characteristic function of a subset of K described by a single polynomial inequality

g D KS S D fz 2 CI pz z gt 0g

Moreover the degree of the defining polynomial p is in this case less than N in zand in z For details see for instance [67] Thus there is no surprise that basic semi-algebraic sets pop-up when dealing with finitely many moments of a shade function

The moment matrix Mk`N1k`D0 is obviously positive-semidefinite but it fulfills

more subtle positivity constraints A classical way of understanding the latter is tointerpret the moment data in a Hilbert space context and possibly to isolate the ldquofreehidden parametersrdquo from the correlated entries Mk` This was the royal way in onevariable proposed and developed by the founders of Functional Analysis startingwith F Riesz M Riesz M Krein M Stone J von Neumann Following themseveral generations of mathematicians statisticians and engineers refined this classof inverse problems The monographs [3 83 104] stand aside for clarity and depthas the basic references for these topics

When speaking about Hilbert space operators the word ldquoquantizationrdquo popsup as these infinite dimensional objects quantize their observable and geometrictangible spectra A variety of possible Hilbert space quantizations are in use todaythe standard Fock space representation of creation and annihilation operators thepseudo differential calculus in Euclidean space Toeplitz algebras Hankel operatorsto name only a few The present notes are not an exception with one distinctive markto make from the very beginning

We quantize shaded images in two dimensions by linear and bounded Hilbertspace operators T 2 L H subject to the commutation relation

ŒTT D ˝

where ˝ is a non-negative rank-one operator The departure from the canonicalcommutation relation ŒTT D I is notable as we allow T to be bounded Theirreducible part of T containing the vector in its range is classified by the principalfunction g 2 L1C 0 g 1 ae the link between the two being offered by aremarkable determinantal formula discovered half a century ago by Pincus [76]

detT zT wT z1T w1 D

detŒI ˝ T z1T w1 D

1 hT w1 T z1i D

expŒ 1

z N Nw jzj jwj gt kTk

1 Introduction 3

Incidentally the additive analog of the above multiplicative representation knowntoday as Helton-Howe trace formula [58]

traceŒpTT qTT D 1

J p qgdA p q 2 CŒz z

where J stands for the Jacobian of the two polynomials was the source ofA Connesrsquo cyclic cohomology theory [15]

Our mathematical journey starts here The exponential transform

Egzw D expŒ 1

z N Nw

of a compactly supported shade function can be defined for all values of zw 2C C It serves as a polarized potential (indeed a Riesz potential renormalized andevaluated at a critical exponent) its function theoretic properties provide amongother things an elegant treatment of the truncated moment problem alluded to aboveand an insight into the announced skeleton topics We soon depart from grey shadesand focus only on black and white photos that is we consider E˝zw in the specialcase of g D ˝ with ˝ a bounded open subset of C

First are the positivity properties of E˝ The kernel 1E˝zw is positive semi-definite on C C and its spectral factorization hT w1 T z1i defineswithout ambiguity the ldquoquantizedrdquo entity the irreducible hyponormal operator Twith rank-one self-commutator having g D ˝ as principal function Already thissingles out the localized resolvent T z1 which is defined without ambiguityeven across the spectrum of T Moreover the germ at infinity of the pointwiseinverse diagonalizes T instead

E˝zwD 1C hT z1 T w1i jzj jwj gt kTk

The experts will recognize here a very special feature not present in the case ofother natural ldquoquantizationsrdquo of˝ as the multiplication by the complex variable onBergman or Hardy-Smirnov space

The spectral picture of the operator T associated as above to the open set˝ is very simple the spectrum coincides with the closure of ˝ the essentialspectrum is the boundary of ˝ (assuming no slits) and the Fredholm index atevery 2 ˝ is precisely equal to one Moreover in this case T is injectiveand dim kerT D 1 It is precisely the fact that T has many eigenvectorswhich makes the exponential transform viewed as a potential of the uniform masssupported on˝ more appropriate for inverse balayage In this direction we mentionthat the resolvent of T localized at the vector extends analytically across ˝as far as the Cauchy transform of ˝ extends When combined with the explicitfactorization of 1 E˝ in terms of this resolvent one obtains a novel proof ofSakairsquos regularity of free boundaries [36]

4 1 Introduction

Geometrically for analytic boundaries ˝ the kernel E˝zw can be intrinsi-cally defined as a solution of a Riemann-Hilbert factorization problem connectingE˝ to its canonical Schwarz reflections in each variable The relationship betweenthe exponential transform and Schwarz function Sz D z z 2 ˝ performing thereflection z 7 Sz in the boundary of ˝ is rather deep For instance the case ofquadrature domains that is when Sz is a meromorphic function in ˝ is detectedand characterized by the rationality of the exponential transform

E˝zw D Qzw

PzPw Q 2 CŒz z P 2 CŒz

In addition the quantized object adds a convenient matrix degeneracy to the picture˝ is a quadrature domain if and only if the linear span H1 of the vectors Tk k 0 is finite dimensional In this case the denominator Pz entering in polarized formin the denominator of E˝zw is the minimal polynomial of TjH1

Addingto these observation the remarkable fact that sequences of quadrature domainsapproximate in Hausdorff metric every planar domain we gain a precious insightinto a dense non-generic class yet finitely determined algebraicmatricial objectscentral for our study The topics of quadrature domains has reached maturity havingquite a few ramifications to fluid mechanics potential theory integrable systemsand statistical mechanics The collection of articles [19] offers an informative andaccessible panorama of the theory of quadrature domains

The approximation scheme we pursue in the present notes is a Galerkin methodalong Krylov subspaces of the hyponormal operator T associated to a planar domainor shade function g Specifically one starts with the non-decreasing sequence offinite dimensional subspaces Hn D spanfTk W 0 k lt ng The compression Tn ofT to Hn has its spectrum identical to the zero set of the orthogonal polynomial pn ofdegree n derived from the inner product

p q WD h pT qTi p q 2 CŒz

Equivalently we can invoke a Padeacute approximation scheme applied to the exponen-tial transform which identifies in a remarkable algebraic identity these objects as aformal series at infinity

E˝zw D 1 hTn w1 T

n z1i C Rnzw

with the residual series Rnzw involving only monomials zk1w`1 in the rangek ` D NN or maxk ` gt N There is no surprise that the denominator of therational approximate is precisely pnzpnw while the numerator is a rather specialpolynomial a weighted sum of hermitian squares of the form

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

with deg qj D j for all j in the range 0 j lt n The algebraic curve given byequation jpnzj2 D Pn1

jD0 jqjzj2 is called a generalized lemniscate and carriesa distinctive (real) geometry The cluster set of the spectra of the finite centraltruncations Tn make up a certain ldquoquantized ridgerdquo The approximation scheme isstationary that is Hn D HN n N for some positive integer N if and only if g isthe characteristic function of a quadrature domain

The comparison with well studied similar approximation procedures eitherrelated to complex orthogonal polynomials Padeacute approximation in several vari-ables random matrix theory or finite central truncations of structured infinitematrices is in order In contrast to these frameworks in our case the inner productis not derived from a Lebesgue space hence the well established techniques ofpotential theory prevalent in the quasi-totality of orthogonal polynomial studiesare not available A welcome substitute is provided by the theory of hyponormaloperators which implies for instance that the zeros of the orthogonal polynomialspn are contained in the convex hull of the support of the shade function g Alsowe can prove by Hilbert space methods a rigidity result stating that the orthogonalpolynomials pn satisfy a three term relation if and only if g is the characteristicfunction of an ellipse

In the present work we focus on bounded planar sets ˝ with real algebraicboundary In this case the exponential transform and its analytic continuation fromthe exterior region provides a canonical defining function of the boundary

˝ D fz 2 C E˝z z D 0g

Moreover the rate of decay of E˝z z towards smooth parts of the boundary isasymptotically equivalent to distz ˝ qualifying the exponential transform asthe correct potential to invoke in image reconstruction The reconstruction algorithmwas already exposed in [49] and is not reproduced in the present notes The analyticextension properties of the localized resolvent T z1 and hence of theexponential transform is driven by the analytic continuation of the Schwarz functionof the boundary As a consequence the quantized ridge tends to lie deep inside theoriginal set ˝ A good third of our essay is devoted to a dozen of examples andnumerical experiments supporting such a behavior On the theoretical side we provea striking departure from the theory of complex orthogonal polynomials in Bergmanor Hardy-Smirnov spaces the quantized skeleton does not ldquoseerdquo external disks (ormore general external quadrature domains) The Glossary at the end of these notescontains a list of standard and ad-hoc notations

Chapter 2The Exponential Transform

Abstract The basic definitions of the exponential transform E˝zw of a planardomain ˝ C and various functions derived from it are recorded in this chapterThe exponential moments having the exponential transform as the generatingfunction are introduced In addition several positivity properties of the exponentialtransform are established Finally it is shown that the exponential transform asa function of z with w kept fixed can be characterized as being (part of) theunique holomorphic section of a certain line bundle over the Riemann sphere takinga prescribed value at infinity or alternatively as being the unique solution of acorresponding Riemann-Hilbert problem

21 Basic Definitions

Definition and many elementary properties of the exponential transform are listedin [36] We recall some of them here

Definition 21 The exponential transform of a function g 2 L1C 0 g 1 isthe function defined at every point zw 2 C C by

Egzw D exp Œ 1

z N Nw (21)

We shall almost exclusively work with the case that g is the characteristic functionof a bounded open set ˝ C and then we write E˝zw in place of E˝ zw Ifthe set ˝ is obvious from the context we may delete it from notation The singularintegral in the above definition may diverge for certain points on the diagonalz D w D a in this case we take Ega a D 0 by convention We note that thisregularization turns Eg into a separately continuous function on the whole C C

The exponential transform E˝zw is analytic in z anti-analytic in w when thesevariables belong to the complement of ˝ henceforth denoted˝e whereas in otherparts of space it has a mixed behavior We shall sometimes denote the exterior partof the exponential transform as well as possible analytic continuations of it across˝ by Fzw

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

8 2 The Exponential Transform

In the remaining parts of CC the following functions are analytic (more preciselyanalytic in z anti-analytic in w)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Note that Gzw D Gw z The same letters F G G H will be used for theanalytic continuations of these functions across ˝ when such continuations existThis occurs whenever ˝ consists of smooth real analytic curves

The behavior at infinity is

Ezw D 1 j˝jz Nw C Ojzj C jwj3 jzj jwj 1 (26)

Ezw D 1 C˝z

Nw C Ojwj2 jwj 1 (27)

C˝z D 1

z^ d N (28)

is the Cauchy transform of the characteristic function of ˝ Note that the expo-nential transform Ezw simply is the exponential of the double Cauchy transform

C˝zw D 1

z^ d N

N Nw (29)

This is a locally integrable function on C C despite C˝z z D 1 for z D w 2˝ (and for most points with z D w 2 ˝) For more general functions g than thecharacteristic function of an open set we write the Cauchy transform as

Cgz D 1

It satisfies Cg=Nz D g in the sense of distributions and this property was used forthe second equalities in (23)ndash(25) (the first equalities there are just definitions)

Taking distributional derivatives of C˝zw gives at least under some lightregularity assumption on ˝

2C˝zw

NzwD ız w˝z˝w zw 2 C (210)

21 Basic Definitions 9

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

By combining this with the asymptotic behavior (26) one can represent 1 Ezwas a double Cauchy integral

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

The Cauchy and exponential transforms can also be written as boundary integralsin various ways For example the exterior and interior functions Fzw andHzw respectively can be expressed by the same formula

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

The proof is straight-forward A further remark is that the double Cauchy transformcan be viewed as a Dirichlet integral specifically its real part can be written as

Re C˝zw D 1

d log j zj ^ d log j wj

where the star is the Hodge star (dx D dy dy D dx) This gives an interpretationto the modulus of the exponential transform in terms of a mutual energy for ˝provided with uniform charge distribution

Example 21 For the disk ˝ D DaR the analytic pieces of the exponentialtransform are

Fzw D 1 R2

z a Nw Na zw 2 DaRe DaRe

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

R2 z a Nw Na zw 2 DaR DaR

Clearly the functions F G G here admit analytic continuations so that the variablewhich is in DaRe extends its range to all of C n fag

For later use we record the following formula an immediate consequence of thedefinition of Ezw and (25) if ˝1 ˝2 then

H˝1zwE˝2n˝1zw D H˝2zw for zw 2 ˝1 (213)

Combining with Example 21 we see that Hzw is essentially one over theexponential transform of the exterior domain In fact applying (213) to theinclusion˝ D0R for R sufficiently large gives

H˝zw D 1

R2 z NwED0Rn˝zw (214)

Occasionally we shall need also the four variable exponential transform definedfor any open set ˝ in the Riemann sphere P D C [ f1g by

E˝zwI a b D exp Œ1

a ^ d N

N Nw d NN Nb

D exp ŒC˝zwI a b D E˝zwE˝a b

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

Thus the two variable transform is recovered as Ezw D EzwI 11 Thefour variable transform behaves well under complementation and under Moumlbiustransformations Indeed denoting by z W a W w W b the classical cross ratio (see[2]) we have

Lemma 21 For any open set ˝ P

E˝zwI a bEPn˝zwI a b D EPzwI a b

EPzwI a b D jz W a W w W bj2 D ˇ z wa b

z ba w

And for any Moumlbius map f we have

Ef ˝ f z f wI f a f b D E˝zwI a b (216)

Similarly for C˝zwI a b

22 Moments 11

Proof The first statement is obvious from the definition (215) To establish therelationship to the cross ratio one may use that the two variable exponentialtransform for a large disk D0R is (inside the disk)

ED0Rzw D jz wj2R2 z Nw zw 2 D0R

as is obtained from (25) together with Example 21 then insert this into (215) andfinally let R 1

To prove (216) one just makes the obvious variable transformation in theintegral Then the required invariance follows from the identity

f 0df f z

f 0df f a

which is a consequence of both members being Abelian differentials on P withexactly the same poles (of order one) and residues

The properties described in Lemma 21 indicate that the four variable exponentialtransform is a natural object to consider The two variable transform behaves in amore complicated way under Moumlbius maps And under other conformal maps thanMoumlbius maps there seems to be no good behavior for any version of the exponentialtransform

22 Moments

The following sets of moments will enter our discussions

bull The complex moments

Mkj D 1

zkNzjdAz D zk zjL2˝

(k j 0) make up the Gram matrix leading to the Bergman orthogonalpolynomials on˝

bull The harmonic (or analytic) moments are

Mk D Mk0 D 1

bull The exponential moments Bkj (sometimes to be denoted bkj) are defined by

zkC1 NwjC1 D 1 exp ŒXkj0

zkC1 NwjC1 (217)

and they make up the Gram matrix for what we will call exponential orthogonalpolynomials In Chap 3 we shall introduce an inner product h i D h iH ˝

such that Bkj D hzk zji It is easy to see that Bk0 D Mk0 D Mk for all k 0

Computation of the exponential moments for 0 k j N requires onlyknowledge of the complex moments for the same set of indices However for this towork out in a numerically efficient way it is necessary to use a recursive procedureThe following effective procedure was suggested to us by Roger Barnard

Write (217) briefly at the level of formal power series

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

and inserting the power series and equating the coefficients for 1=zkC2 NwjC1 gives therecursion formula

k C 1Mkj Bkj DXpq

p C 1MpqBkp1jq1 k j 0

where the sum ranges over 0 p lt k 0 q lt j This gives Bkj efficiently fromMpq with 0 p k 0 q j and the previously computed Bpq with 0 p lt k0 q lt j

Note that B above is simply the power series expansion of 1 Ezw at infinityand M is similarly the power series of (minus) the double Cauchy transform (29)In summary the generating functions for the moments are

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

23 Positive Definiteness Properties 13

23 Positive Definiteness Properties

As is known from the theory of hyponormal operators see [74] and is proveddirectly in [39] the exponential transform enjoys a number of positivity propertiesWe recall them here in slightly extended forms and with sketches of proofs

Lemma 22 For arbitrary finite sequences of triples zk ak k 2 P P C wehave

C˝zk zjI ak ajkNj 0 (218)

with C1 as an allowed value for the left member If the sets fzkg and fakg aredisjoint and no repetitions occur among the zk then we have strict inequality gt 0unless all the k are zero

Assuming that the left member in (218) is finite we also have

E˝zk zjI ak aj 0

with the same remark as above on strict inequality

Proof We have

C˝zk zjI ak ajkj D 1

akj2 dA 0

which proves (218) The statement about strict inequality also follows immediatelyAs a consequence of Schurrsquos theorem the exponential of a positive semidefinite

matrix is again positive semidefinite (see [18] for example) Therefore

E˝zk zjI ak ajDXkj

exp ŒC˝zk zjI ak ajkj 0

under the stated assumptionsFrom the above we conclude the following for the two variable transforms

Lemma 23 For any bounded open set ˝ C the following hold

(i) C˝zw is positive definite for zw 2 ˝e(ii) 1

Ezw is positive definite for zw 2 ˝e

(iii) Hzw is positive definite for zw 2 ˝(iv) 1 Ezw is positive semidefinite for zw 2 C(v) 1

Ezw 1 is positive semidefinite for zw 2 ˝e

Proof i and ii follow immediately from the previous lemma by choosing ak D1 for all k To prove iii we choose some R gt 0 such that ˝ D0R anduse (214) Here the factor

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

is clearly positive definite while the other factor 1=ED0Rn˝zw is positivesemidefinite by ii Hence iii follows on using Schurrsquos theorem Next iv followsfrom iii by means of the representation (212) and v follows from ii and ivtogether with Schurrsquos theorem

Remark 21 Whenever one has a positive definite function like Hzw in iii ofthe lemma then one can define a Hilbert space either by using it directly to definean inner product or by using it as a reproducing kernel These two methods canactually be identified via the linear map having Hzw as an integral kernel

Since much of the text will be based on these ideas we explain here theunderlying philosophy using sums instead of integrals for simplicity of expositionLet the kernel in general be Kzw analytic in z anti-analytic in w Then oneHilbert space is generated by finite sums

Pj ˛jızj with the inner product is defined

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

In the case Kzw D Hzw this will be our main Hilbert space to be denotedH ˝

This Hilbert space is thus generated by singular objects (Dirac distributions) andcan be rather strange However as Kzw is a quite regular function a much nicerHilbert space consisting of anti-analytic functions is obtained via the identificationmap (one-to-one by the assumed positive definiteness)

˛jızj 7X

˛jKzj

for which the same inner product is kept ie

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

We have indexed the inner product by RK because the space now becomes areproducing kernel Hilbert space with in fact Kzw as the reproducing kernelsetting ˚ D P

j ˛jKzj and letting the second factor be just Kw we have

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

The above will be applied for the choice iii in Lemma 23 Kzw D HzwBut also the other choices in the lemma are interesting in particular for spacesof analytic functions i is related to the ordinary Bergman space see (327) iiand v give an inner product considered by Pincus-Xia-Xia [77] see (329) andSect 37 while iv gives the same as iii but with the inner product written in adifferent way see (328) and also (33)

Next we specialize to comparison with disks and half-planes

Lemma 24 Some specific positivity assertions are

(i) If ˝ D0R then R2 z NwH˝zw is positive definite for zw 2 ˝ (ii) If ˝ D0 r D then 1 r2

z Nw H˝zw is positive definite for zw 2 ˝ (iii) Assume that ˝ D where D is an open half-plane let a 2 D and let b D a

be the reflected point with respect to D Then

Nw NaNw Nb H˝zw zw 2 ˝

is positive definite

Proof For i we use that (by (213) and ii in Lemma 23)

HD0Rzw H˝zw D 1

ED0Rn˝zw

is positive definite for zw 2 ˝ and then insert HD0R D 1R2z Nw (see Example 21)

For ii we similarly use that

H˝zwED0rzw D H˝[D0rzw

is positive definite for zw 2 ˝ and insert ED0r D 1 r2

z Nw Finally for iii we use the formula (216) for how the four variable exponential

transform changes under a Moumlbius map f We take this to be

b (219)

which maps the half plane D onto the unit disk in particular f ˝ D Using that

H˝zw D E˝zw

jz wj2 D E˝zwI b bE˝z bE˝bw

jz wj2 E˝b b

by (25) (215) we then obtain

Nw NaNw Nb H˝zw

D 1 f zf w Ef ˝ f z f wI f b f b

jf z f wj2 ˇ f z f w

ˇ2 E˝z bE˝bw

E˝b b

D 1 f zf w Hf ˝ f z f w ja bj2E˝z bE˝bwjz bj2jw bj2E˝b b

Here the last factor (the quotient) is of the form czw with c gt 0 hence ispositive definite and also the first factor is positive definite because it equals

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Thus part iii of the lemma follows

24 The Exponential Transform as a Section of a Line Bundle

In this section we assume that ˝ is fully real analytic with Schwarz function (see[16 101]) denoted Sz Thus Sz is analytic in a neighborhood of ˝ and satisfies

Sz D Nz z 2 ˝ (220)

The functions F G G H defined in terms of E by (22)ndash(25) are analyticin their domains of definition have analytic continuations across the boundariesand satisfy certain matching conditions on these boundaries Taking the analyticcontinuations into account the domains of definition cover all of P P and thematching conditions can be formulated as transition formulas defining holomorphicsections of certain line bundles These sections will then be uniquely determined bytheir behaviors at infinity This gives a new way of characterizing the exponentialtransform In an alternative and essentially equivalent language the exponentialtransform can be viewed as the unique solution of a certain Riemann-Hilbertproblem Below we shall make the above statements precise

24 The Exponential Transform as a Section of a Line Bundle 17

Let w 2 ˝e be fixed Then by the definitions (22) (23) (220) of F G and S

GzwSz Nw D Fzw (221)

for z 2 ˝ Here Gzw remains analytic as a function of z in some neighborhoodof ˝ say in U ˝ and Fzw similarly remains analytic for z in say V ˝cWith U V chosen appropriately Sz is analytic in U V and (221) remains validthere At the point of infinity (for z) we have F1w D 1

We interpret the above as saying that the pair GwFw represents asection of the line bundle over P defined by the transition function Sz Nwz 2 U V The Chern class of this line bundle is

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Recall that the Chern class equals the difference between the number of zeros andthe number of poles for an arbitrary meromorphic section (not identically zero) andthat a line bundle on P is uniquely determined by its Chern class Having Chern classzero therefore means that the bundle is equivalent to the trivial bundle (transitionfunctions identically one) as is also evident from the way we started namely byhaving a non-vanishing holomorphic section It follows that the dimension of thespace of holomorphic sections equals one so any holomorphic section is determinedby its value at one point Therefore the holomorphic section GwFw isuniquely determined by the property that it takes the value one at infinity

With w 2 ˝ a similar discussion applies to the pair HwGw Bydefinition of G and H we then have the transition relation

HzwSz Nw D Gzw (222)

thus with the same transition function Sz Nw as above However in the presentsituation the Chern class is C1 because w now is inside ˝ in the integrals aboveand the bundle is therefore equivalent to the hyperplane section bundle See [109][29] for the terminology Thus any nontrivial holomorphic section has exactly onezero and is uniquely determined by the value of its derivative at this zero Theparticular section HwGw vanishes at infinity with the expansion there

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

as follows from the behavior of Ezw at infinity In particular the sectionHwGw is uniquely determined solely by these properties

As a side remark the functions Gzw Gzw are not only analytic across˝ ˝ they are also non-vanishing (except at the point of infinity as exhibitedfor Gzw in (223)) Therefore the relations (221) and (222) give preciseinformation on the nature of the zeros of Fzw on ˝˝ and the corresponding

singularities of Hzw there Note that 1=Hzw (which is similar to Fzw for atruncated exterior domain) is still holomorphic across ˝ ˝

We summarize the above discussion

Theorem 21 Assume that ˝ is analytic and let w 2 C n ˝ be fixed Then thefunction Sz Nw which is holomorphic and non-vanishing in a neighborhoodof ˝ defines a holomorphic line bundle w on P with respect to the covering˝ neighborhoodP n˝(i) If w 2 ˝e then w has Chern class zero hence it has a unique holomorphic

and non-vanishing section which takes the value one at infinity This section isrepresented by the pair GwFw

(ii) If w 2 ˝ then w has Chern class one hence it has a unique holomorphicsection which vanishes at infinity with derivative one there (ie behavior z1COz2 z 1) and has no other zeros This section is represented by thepair HwGw

There is also a limiting version of the above for w 1 See Proposition 21below

One may extend the above considerations by treating the two variables z andw jointly However in this case there will be a singular set namely the one-dimensional set D fzw 2 C2 W z D w 2 ˝g In fact in a neighborhoodof we have the relation

Fzw D Sz Nwz SwHzw (224)

but on itself the transition function Sz Nwz Sw thought to define a linebundle vanishes while Hzw becomes infinite Anyhow Fzw is well-behaved(analyticantianalytic) in a full neighborhood of The restricted conclusion will bethat the quadruple FGGH defines a holomorphicantiholomorphic sectionof a line bundle just in P P n

25 A Riemann-Hilbert Problem

We can interpret the factorization at interface formulas obtained so far as a Riemann-Hilbert problem with relations to hold on ˝ and with Nz in place of Sz

GzwNz Nw D Fzw z 2 ˝ w 2 ˝e (225)

HzwNz Nw D Gzw z 2 ˝ w 2 ˝ (226)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

25 A Riemann-Hilbert Problem 19

This requires less regularity of ˝ and may therefore be convenient from somepoints of view The requirements at infinity are as before

F1w D Fz1 D 1 (229)

Gzw D 1Nw C Ojwj2 Gzw D 1z

C Ojzj2 (230)

A particular consequence of the last transition relation and (230) is thatZ˝

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

After turning the first integral to an area integral this gives

HzwdAw D 1 z 2 ˝ (231)

a formula which will be needed later onThe decomposition (22)ndash(25) of Ezw into analytic pieces can be extended

to the four variable transform EzwI a b One practical way to formulate such astatement is to say that the function

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

is analytic in each component of P n ˝4 Here denotes the characteristicfunction of ˝

D(1 2 ˝0 hellip ˝

Cf Theorem 2 in [47]The relationships between the functions F G G and H can also be expressed

by means of the Cauchy transform To this end we make the following observation

Lemma 25 Assume f and g are holomorphic in ˝ h is holomorphic in ˝e allhaving continuous extensions to ˝ and that

(Nzf zC gz D hz z 2 ˝hz 0 z 1

Then the combined function

(Nzf zC gz z 2 ˝hz z 2 ˝e

is identical with the Cauchy transform of f more precisely of the function

(f z z 2 ˝0 z 2 ˝e

Conversely for f holomorphic in˝ and continuous up to ˝ the Cauchy transformof (234) is of the form (233) with (232) holding

Proof This is immediate from basic properties of the Cauchy transformIn the notations and assumptions of the lemma we may consider the Cauchy

transform (in a restricted sense) simply as the map f 7 h in particular as a maptaking analytic functions in˝ to analytic functions in˝e As such a map it connectsin our context the different pieces F G G H of the exponential transform

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

where w 2 ˝ is considered as a parameter we get

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

Example 23 With w 2 ˝e as parameter and

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

CGwz D Fzw 1 z 2 ˝e

Example 24 By taking the Cauchy transform in both variables one can pass directlyfrom Hzw to Fzw or even from Hzw to the complete transform Ezw by

means of Eq (212) Similarly by considering the asymptotics at w D 1 or bycombining (28) with (231)

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

In addition using (235) one finds that

C˝z D 1

Gzw dAw z 2 ˝e (237)

As a final remark we mention that not only the exponential transform (ie theexponential of the double Cauchy transform (29)) but also the exponential of theone variable Cauchy transform (28) can be viewed as (part of) a section of a linebundle In fact taking f z D 1 in (232) gives hz D C˝z for z 2 ˝e andreplacing Nz by Sz on ˝ renders (232) on the form

eSz egz D eC˝z (238)

which by analytic continuation holds for z in a neighborhood of ˝ In (238) egz

is just some non-vanishing holomorphic function in ˝ which we do not care muchabout while eC˝z is holomorphic and non-vanishing in ˝e regular at infinity andtaking the value one there

Now we may view eSz as the transition function for a line bundle on P thenhaving Chern class zero It follows that there is a unique holomorphic cross sectionof that bundle which takes the value one at infinity And (238) says that therestriction to ˝e of that section is exactly the exponential of the Cauchy transformof ˝ In summary

Proposition 21 With ˝ real analytic eSz defines a holomorphic line bundle onP with Chern class zero Its unique non-vanishing holomorphic section which takesthe value one at infinity is represented in ˝e by eC˝z

Chapter 3Hilbert Space Factorization

Abstract The positivity properties of the exponential transform define in a canon-ical way a Hilbert space H ˝ and a (co)hyponormal operator acting on it suchthat the exponential transform itself appears as a polarized compressed resolvent ofthis operator There are many variants of this procedure but they are all equivalentHistorically the process actually went in the opposite direction starting with anabstract Hilbert space and hyponormal operator with rank one self-commutatorthe exponential transform arose as a natural characteristic function obtained asthe determinant of the multiplicative commutator of the resolvent If one considersH ˝ as a function space the analytic functions in it are very weak ie have asmall norm and for a special class of domains the quadrature domains the analyticsubspace Ha˝ even collapses to a finite dimensional space Some more generalkinds of quadrature domains are discussed in terms of analytic functionals and wealso show that some integral operators based on the exponential transform can beinterpreted in terms of Silva-Koumlthe-Grothendieck duality

31 Definitions and Generalities

In the sequel we assume that Hzw is integrable

jHzwjdAzdAw lt 1 (31)

We do not know whether this always holds but as shown in [39] (appendix there) itholds at least when ˝ is Lipschitz in particular when it is piece-wise real analyticIf ˝ is fully real analytic one even has

jHzwj2dAzdAw lt 1 (32)

see again [39]In view of the positivity properties proved in Sect 23 one can define a positive

semi-definite Hermitian form on the set DC of smooth test functions with compact

24 3 Hilbert Space Factorization

support in C by

h f gi D 1

1 EzwNf zgwdAzdAw (33)

1 Ezwd f zdzdgwdw f g 2 DC

We are following [38 39] here Several other choices of inner product are in usesee [74 77] as well as Sects 34 and 37 below

The second expression above indicates that it is natural to consider fdz and gdzas differential forms and as such they need not have compact support it is enoughthat they are smooth also at infinity (ie smooth on all P) Upon partial integrationthe above formula becomes

h f gi D 1

Hzwf zgwdAzdAw (34)

hence h f gi depends only on the values of f and g in ˝ By (31) all functionsf g 2 L1˝ can be allowed in (34) and we have the estimate

jj f jj Cjj f jj1˝ (35)

where jj f jj2 D h f f i Also distributions and analytic functionals with compactsupport in ˝ can be allowed in (34)

Despite Hzw being positive definite there are functions f 2 L1˝ for whichjj f jj D 0 for example any function of the form f D N with a test function withcompact support in ˝ Therefore to produce a Hilbert space having h i as innerproduct one first has to form a quotient space modulo the null vectors and thentake the completion of this The resulting Hilbert space will be denoted H ˝ It isalways infinite dimensional and separable When necessary we will write the innerproduct defined by (34) as h f gi D h f giH ˝ and similarly for the norm

The construction above gives a natural map taking functions to their equivalenceclasses

˛ W L1˝ H ˝ (36)

This map has (by definition) dense range but is not injective If ˝ is analytic sothat (32) holds (henceforth assumed) ˛ extends to

˛ W L2˝ H ˝

The adjoint operator ˛0 goes the opposite way between the dual spaces

˛0 W H ˝0 L2˝0

and is automatically injective (because ˛ has dense range)

31 Definitions and Generalities 25

Keeping the assumption (32) the adjoint map ˛0 has some concrete manifesta-tions For example representing functionals in the above Hilbert spaces by innerproducts it gives rise to a map

ˇ W H ˝ L2˝

which is bounded and injective Precomposing it with ˛ gives the operator

H D ˇ ı ˛ W L2˝ L2˝

We name it H because it has an explicit presentation as an integral operator withkernel Hzw

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

By construction Hf only depends on the equivalence class of f 2 L2˝ in H ˝ie on ˛ f and we have

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

It follows that H is a positive operator and it is also seen that its range consists onlyof anti-analytic functions in particular it is far from being surjective The norm ofH D ˇ ı ˛ as an operator is bounded above by jjHjjL2˝˝ and it also follows thatjj˛jj D jjˇjj pjjHjjL2˝˝

As indicated in Remark 21 the one-to-one image of H ˝ under the map ˇis if the inner product in H ˝ is kept in ˇH ˝ a reproducing kernel Hilbertspace of anti-analytic functions having Hzw as the reproducing kernel

Expanding (212) for large z and w gives

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

Since on the other hand

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1 D 1 expŒXkj0

zkC1 NwjC1

this confirms that the exponential moments defined by (217) make up the Grammatrix for the monomials in H ˝

Bkj D hzk zjiH ˝

For future needs we record here the following consequence of (231)

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Strictly speaking the left member here should have been written h˛h ˛1i butquite often we will consider ˛ as an identification map and suppress it from notationThe equivalence class in H ˝ of the constant function one will have a special rolelater on and we write it in bold ˛1 D 1

32 Restrictions and Extensions

The operator ˛ has a very big kernel and therefore it is not reasonable to considerH ˝ as a function space However the operator ˇ canonically embeds H ˝

in a space of anti-analytic functions in ˝ and as we shall see in Sect 36 formost choices of ˝ the map ˛ embeds the functions analytic in a neighborhood of˝ in a dense subspace of H ˝ However the norm in H ˝ is much weakerthan traditional norms on spaces of analytic functions and there will usually be nocontinuous point evaluations for example

A sign that H ˝ is not a good function space is that on considering aninclusion pair ˝1 ˝2 there is no continuous restriction operator H ˝2 H ˝1

Example 31 Consider the inclusion D1 1 D 2 where 0 lt lt 1 Ingeneral if f is analytic and bounded in DaR we have (writing just f in placeof ˛ f ) jj f jjH DaR D Rj f aj Here we used Example 21 and the mean-valueproperty of analytic functions Thus for fnz D zn

jj fnjjH D11 D 1 jj fnjjH D2 D 2n

hence there is no constant C lt 1 such that jj fnjjH D11 Cjj fnjjH D2 for alln So at least the most naive way of trying to define a bounded restriction operatorH D 2 H D1 1 does not work

On the other hand it is possible to define extension operators with bounded norm(actually norm one) In fact if ˝1 ˝2 and f 2 L1˝1 then on extending f byzero on ˝2 n˝1 we have

jj f jjH ˝2 jj f jjH ˝1

33 Linear Operators on H ˝ 27

This is an immediate consequence of (213) and iv of Lemma 23 the latter appliedto 1 E˝2n˝1

33 Linear Operators onH ˝

Let Z be the ldquoshiftrdquo operator ie multiplication by the independent variable z onH ˝

Z W H ˝ H ˝ Zf z D zf z (310)

This is a bounded linear operator in fact its norm is

jjZjj D supfjzj W z 2 ˝gHere the upper bound (ie the inequality ) follows from i of Lemma 24 whichimmediately gives the desired inequalityZ˝

Hzwzf zwf wdAzdAw R2Z˝

Hzwf zf wdAzdAw

The lower bound follows by choosing f to have all its support close to a point z 2 ˝at which jzj nearly equals its supremum For example if Da ˝ then f DDa gives jj f jj D 2

pHa a jjZf jj D a2

If ˝ D0 r D then the inverse Z1f z D 1z f z exists and has norm

jjZ1jj 1r by ii of Lemma 24 It is easy to see that equality holds if r is chosen

largest possible hence jjZ1jj D 1=dist0˝ More generally if a hellip clos˝ thenZ a1 exists and has norm 1=dist a˝ Thus denoting by Z the spectrumof Z we have Z clos˝ In fact equality holds

Z D clos˝ (311)

By Z we denote the operator

Zf z D Nzf z (312)

by Z the adjoint of Z and by C the Cauchy transform considered as a linearoperator H ˝ H ˝

Cgz D Cgz D 1

z z 2 ˝ (313)

Finally 1 ˝ 1 denotes the operator

1 ˝ 1 W h 7 hh 1i1

which is a positive multiple of the orthogonal projection onto the one-dimensionalsubspace spanned by 1 (the function identically equal to one)

Proposition 31 The so defined operators are bounded on H ˝ and are relatedby

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

In particular Z is cohyponormal ie ŒZZ 0

Proof Using partial integration with respect to w plus the Riemann-Hilbert rela-tion (228) we have

hzf z gzi h f z Nzgzi D 1

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwz wf zCgwd NwdAzC

Hzwf zCgwdAzdAw

Gzwf zCgwd NwdAzC h f zCgzi

Here the boundary integral in the first term vanishes because the integrand is anti-analytic with respect to w in ˝e and tends to zero as 1=jwj2 when jwj 1 inview of the asymptotics of Gzw (see (230)) and Cgw Thus we end up with theidentity

hzf z gzi D h f z Nzgzi C h f zCgzi (316)

This says that

hZf gi D h f Z C Cgi

ie we have proved (314) and thereby also that C is bounded (since Z obviouslyis bounded like Z) We emphasize that we did not use the definition of Hzw

directly in the above computation only its property of being part of a section (anti-holomorphic section with respect to w to be precise) vanishing at infinity of a certainline bundle equivalently the solution of a Riemann-Hilbert problem

Next we compute the commutator ŒZC D ZC CZ

ŒZC f z D z 1

fdA D h f 1i 1 D 1 ˝ 1f z

Since obviously ŒZZ D 0 the remaining statements of the proposition now followExpressing the exponential and Cauchy transforms in terms of the above

operators we have for all zw 2 C

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

These relations follow immediately from (212) (236) and the definition (34) ofthe inner product We may also identify the Cauchy kernel itself as an element inH ˝ as

kz D Z z11 (319)

The exponential moments appear are

Bkj D hZk1Zj1i

We mention next a determinantal formula for E˝zw in terms of Z

E˝zw D detZ NwZ zZ Nw1Z z1 (320)

valid for zw 2 ˝e This is proved in the same way as a similar formula inAppendix A namely by switching the first two factors and using that ŒZ zZ Nw D ŒZZ D 1 ˝ 1 which is a rank one projection Then the determinant comesout to be 1 hZ z11 Z w11i which is Ezw The determinant existsbecause it is of the form detI C K with K a trace-class operator and then thedefinition of the determinant reads

detI C K D expŒtr logI C K D expŒtr1X

In our case the operator K has finite rank (rank one) and in such cases the productrule detAB D det A det B holds for determinants as above This gives thecomplementary formula

E˝zwD detZ zZ NwZ z1Z Nw1 (321)

D 1C hZ Nw11 Z Nz11i

34 A Functional Model for Hyponormal Operators

The operator Z used above is cohyponormal but it is actually more common towork with models which involve hyponormal operators (see Appendix A and [74]in general for such operators) By minor modifications of the previous definitionsone easily arrives at the standard hyponormal operator T in this context The innerproduct is first to be modified to be

hh f gii D 1

1 Ew zf zgwdAzdAw (322)

Hw zf zgwdAzdAw D hNf NgiH ˝

This gives rise to a Hilbert space which is equivalent to H ˝ in the sense thatconjugation f 7 Nf is an R-linear isometry between the two spaces In the newHilbert space we define the shifts Z Z and the Cauchy transform C as before by theformulas (310) (312) (313) and in addition we set

Cf z D 1

f dAN Nz z 2 ˝

ie Cf D CNf Then it is straight-forward to check that

hhZ C Cf gii D hh f Zgii

This means that on defining an operator T by

T D Z C C

its adjoint with respect to the new inner product is

35 Summary in Abstract Setting 31

In addition one gets

ŒTT D 1 ˝ 1

in particular T is hyponormal The relations to the Cauchy and exponential transformare

1 E˝zw D hhT Nw11 T Nz11ii (323)

C˝z D hh1T Nz11ii

the exponential moments appear as

Bkj D hhTj1Tk1ii

and the formula corresponding to (321) becomes

E˝zwD detT NwT zT Nw1T z1 (324)

D 1C hhT z11 T w11ii

for zw 2 ˝e See Appendix A for more details and references

35 Summary in Abstract Setting

For future needs we summarize the two functional models used so far with innerproducts connected by (322) in terms of an abstract Hilbert space H with a specialvector 0 curren 2 H as follows (this is the abstract version of 1) We work eitherwith a cohyponormal operator A (replacing Z) in H satisfying

ŒAA D ˝

or a hyponormal operator T satisfying

ŒTT D ˝

In either case it is assumed that the corresponding principal function (see [74]) isof the form g D ˙˝ (minus sign in the cohyponormal case) where ˝ C is abounded open set Using h i for the inner product in whatever case we are workingwith we have the basic relationships to the Cauchy and exponential transforms and

exponential moments given by

1 E˝zw D hA z1 A w1iC˝z D hA z1 i

Bkj D hAkAji

respectively

1 E˝zw D hT Nw1 T Nz1iC˝z D hT Nz1i

Bkj D hTjTk

In addition we have the determinantal formulas

E˝zw D detA NwA zA Nw1A z1

D detT zT NwT z1T Nw1

E˝zwD detA zA NwA z1A Nw1

D detT NwT zT Nw1T z1

36 The Analytic SubspaceHa˝

For any set E C we define

OE D f(germs of) functions holomorphic in some open set containing Eg

with the qualification that two such functions which agree on some neighborhoodof E shall be identified The analytic subspace of Ha˝ of H ˝ may be definedas

Ha˝ D closH ˝˛O˝

The Cauchy kernel kz D 1z clearly is in O˝ hence can be considered as

an element of Ha˝ whenever z 2 ˝e The same for all polynomials Whenrestricting to O˝ the map ˛ is in most cases injective (namely when ˝ is not aquadrature domain see Theorem 31 below) therefore it is reasonable to write justf in place of ˛ f when f is analytic

36 The Analytic Subspace Ha˝ 33

For f g 2 O˝ the inner product can be written as a boundary integral

h f gi D 1

1 Ezwf zgwdzd Nw f g 2 O˝ (325)

This agrees with what is obtained from analytic functional calculus namely onwriting

f Z D 1

If zZ z1 dz

where the path of integration surrounds the spectrum of Z ie the closure of ˝ Inserting (317) into (325) gives

h f gi D h f Z1 gZ1i f g 2 O˝

Translating this into a formula for T D Z and the inner product (322) gives

hh f ı conj g ı conjii D hh f T1 gT1ii f g 2 Oconj˝

where conj means conjugation for example f ı conjz D f Nz Thus in the leftmember one takes the inner product between anti-analytic functions in ˝ For Titself one gets a nicer inner product which is used in the analytic functional modelof J Pincus D Xia JB Xia [77 111] (see also Sect 37 below) It is defined by

h f giPXX D hh f T1 gT1ii D (326)

Ezw 1f zgwdzd Nw f g 2 O˝

where the second equality is a consequence of (324)We finally remark that the ordinary Bergman inner product can be written on the

same form as (325)

f gL2˝ D 1

C˝zwf zgwdzd Nw f g 2 O˝ (327)

This follows by transforming the boundary integrals to area integrals andusing (210) If we rewrite (325) as

h f giHa˝ D 1

eC˝zwf zgwdzd Nw f g 2 O˝ (328)

we see that the difference between the two inner product just amounts to anexponentiation of the weight function This is the same relation as in (217) The

analytic model of Pincus-Xia-Xia can also be put into such a form by writing (326)as

h f giPXX D 1

Here one has to be careful to approach the contour of integration ˝ from outside(the cases (327) and (328) are less sensitive) This is because eC˝zw D 1=Ezw(analytically continued across ˝ as 1=Fzw) has a pole at z D w 2 ˝ and themain contribution to the integral actually comes from the residue at this pole Seemore precisely Sect 37

The above three inner products are quite different (327) is exactly the Bergmanspace inner product (ie L2 with respect to area measure) (329) is similar tothe HardySmirnovSzegouml space inner product (L2 with respect to arc length)while (328) gives the rather brittle space Ha˝

37 The Analytic Model

We have seen that the auxiliary Hilbert space H ˝ provides a natural diagonal-ization of the cohyponormal operators Z and T Here we shall see in some moredetail how the analytic model of Pincus-Xia-Xia [77] diagonalizes directly T on theHardy space of ˝ with the price of an inner space distortion

We assume that the boundary ˝ is smooth and real analytic and recall (seeremarks before Theorem 21) that this means that Ezw analytically extends fromthe exterior across ˝ and we also know that this analytic continuation which wedenote Fzw as before satisfies

F D 0 zF curren 0 2 ˝

The partial derivative appearing here may by identified in terms of the functionsGzw and Gzw in (23) (24) as

zF D G D G 2 ˝ (330)

Let T be the irreducible hyponormal operator with rank-one self-commutatorhaving the characteristic function of ˝ as principal function and let bea collection of Jordan arcs in C n ˝ which are homologous to ˝ We maywork in the functional model H ˝ or in an abstract setting (Sect 35) therewill only be notational differences In notations of Sect 35 the determinantalidentity (324) together with Riesz functional calculus gives the Pincus-Xia-Xia

38 A Formal Comparison to Quantum Field Theory 35

inner product as

h f giPXX D h f T gTi D 1

Ezwdzdw

for any germs of analytic functions f g 2 O˝Fix the system of curves but deform to ˝ so that Ezw curren 0 for z 2

and w 2 In this case Eww D 0 but in terms the analytic continuation Fzwof Ezw the function z 7 1=Fzw has just a simple pole at z D w 2 ˝ Theresidue there equals 1=zFww D 1=Gww (recall (330)) Denoting by asystem of curves homologous to but located just inside ˝ the residue theoremtherefore yields

Fzwdzdw D 1

f wgwd Nw

It is easy to see that d NwiGww is positive and hence equal to jdw

jGwwj so all is all wehave for the squared norm

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

Let H2˝ denote the Hardy space associated to the domain ˝ that is theclosure of O˝ in L2˝ jdzj The above integral decomposition implies that thenorm in the ldquoexoticrdquo space Ha˝ has the structure

k f Tk2 D hN C Kf f i2˝

where N is an invertible normal operator and K is a compact operator The strictpositivity of the norm k f Tk implies then that the Hardy space operator N C K DA is self-adjoint positive and invertible Hence

k f TkH ˝ D kpAf k2˝

38 A Formal Comparison to Quantum Field Theory

A first remark is that the notation 1˝ 1 (or ˝ ) used for an operator for examplein (315) actually involves some misuse of notation since one of the functions 1 isto be considered as an element of the dual space (ie is a functional) In P Diracrsquosbra-[c]-ket notation used in quantum mechanics (ji for vectors hj for covectors)the same object would have been denoted j1i ˝ h1j taking into account also that

in quantum mechanics one puts the conjugation on the lefthand factor in innerproducts

Staying within quantum theory Eq (315) is somewhat reminiscent of thecommutation relation for the annihilation and creation operators A and A sayin quantum field theory (see [31 97] for example) In that theory the right memberwould be the identity operator however which using Diracrsquos notation becomes

ŒAA D1X

jD0j ji ˝ h jj (331)

where f jg is an arbitrary orthonormal basis for the Hilbert space used Thus theannihilation operator is also cohyponormal but is necessarily unbounded Oneadvantage with our rank one self-commutator cohyponormal operator Z is that itis bounded

The inner product in Ha˝ gives when written on the form (328) someassociations to Feynman path integrals in quantum field theory Such integrals alsorepresent inner products interpreted as probability amplitudes for transitions forexample the probability h f tf j i tii that a quantum system initially at time ti in astate j i tii will at a later (final) time tf be observed to be in a state j f tf i Such anintegral may look like

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

where ˚ runs over paths in a configuration space SŒ˚ D Rd4xL Œ˚ denotes

the action along the path ˚ here with L the Lagrangian in a four space-timedimensional context The integration is performed over all possible paths joiningthe given initial and final states with integration measure denoted DŒ˚ See[23 31 62 84 97] for example In string theory and conformal field theory theaction may be a pure Dirichlet integral in two dimensions see [78]

If we compare with the inner product (328) for Ha˝ we see that the doubleCauchy transform C˝zw will have the role of being the action (up to a constantfactor) This is completely natural since C˝zw easily can be written as a Dirichletintegral and be interpreted as an energy (mutual energy actually for z curren w) Theclassical action has dimension energy time (the same as Planckrsquos constant bdquo) butwe have constants to play with The main difference however is that the classicalaction is a real-valued function so that the exponent iSŒ˚ is purely imaginarywhile our C˝zw is truly complex-valued On the other hand inserting Planckrsquosconstant the exponential in (332) becomes i

bdquo SŒ˚ and in some contexts bdquo is allowedto take complex values or at least to approach the classical limit bdquo 0 throughcomplex values (cf [97] Sect 1424) One may alternatively compare (328) withthe partition function in quantum statistical mechanics which is similar to theFeynman integral but having a purely real exponential factor

39 Silva-Koumlthe-Grothendieck Duality 37

The two functions f and g in (328) correspond to the states j i tii and j f tf ieven though these are invisible in the right member of (332) They are hidden inthe boundary conditions for the integral but can be restored explicitly as is doneeg in [84] Chaps 5 and 9 One may also represent the time evolution from ti to tfexplicitly by inserting the appropriate unitary operator say eitf ti OH if the systemis governed by a time-independent Hamiltonian operator OH Then the left memberof (332) turns into h f jeitf ti OHj ii0 the subscript zero indicating that we nowhave a different inner product In our context this step could be compared with theinsertion of the operator H in (37) giving an L2-inner product

The difficulty with Feynman integrals is that the integration is taken over aninfinite dimensional space of functions ˚ In practice one has to resort to finitedimensional approximations or perhaps to restrict to a class of functions dependingon finitely many parameters The latter is exactly what is the case in our innerproduct (328) where essentially only the functions

˚ D log z 2 ˝

parametrized by z 2 ˝ appear after having written the integral as a Dirichletintegral The differential DŒ˚ reduces to just dz d Nw with integration along ˝˝

So without claiming any deeper connections we can at least say that there aresome formal similarities between the inner product in Ha˝ and Feynman pathintegrals

39 Silva-Koumlthe-Grothendieck Duality

For some further representations of the inner product we introduce the integraloperator G with kernel Gzw

Gf w D 1

Gzwf zdAz w 2 ˝e f 2 O˝ (333)

This operator takes any f 2 O˝ to an anti-analytic function Gf in ˝e vanishingat infinity This means that Gf 2 O˝e0 where O˝e0 denotes the space offunctions holomorphic in ˝e D P n ˝ and vanishing at infinity Recall next thatthere is a natural duality between O˝ and O˝e0 defined via the pairing

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

which has been studied in depth by Silva [98] Koumlthe [66] Grothendieck [30] Tomake (334) more invariant one could view it as a pairing between the one-form fdzand the function g In any case f is holomorphic in some neighborhood of ˝ and

the integral should be moved slightly into ˝e to make sense The duality statementis that f gduality is a nondegenerate pairing between O˝ and O˝e0 whichexhibits each of these spaces as the dual space of the other

By the definition (23) of the kernel Gzw we may write (333) as

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

On using (325) this gives a representation of the inner product in Ha˝ as

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

Thus in terms of the Silva-Koumlthe-Grothendieck pairing

h f giHa˝ D f Ggduality (335)

A special consequence of (335) is that Gf only depends on the equivalence class off in Ha˝ and then that the map Ha˝ O˝e0 f 7 Gf is injective Therepresentation (335) may be compared with (38) which we may write simply as

h f giHa˝ D Hf gL2˝ D f HgL2˝ f g 2 L2˝

Example 32 Taking f D 1 in (333) gives using (237)

G1w D C˝w w 2 ˝e

Compare with the identity obtained from (231)

H1 D 1

Let now fpng (n D 0 1 2 ) be an orthonormal basis of Ha˝ so thathpk pji D ıkj Then using the above operator G we obtain a basis which is dualto fpng with respect to the pairing duality in (334) by setting

qn D Gpn (336)

Thus the qn are analytic functions in P n˝ vanishing at infinity and satisfying

pkzqjzdz D ıkj

The minus sign can be avoided by replacing ˝ by P n˝

This dual basis fqng can also be identified as the Fourier coefficients of theCauchy kernel

kz D 1

z 2 ˝

where z 2 ˝e is to be regarded as a parameter Indeed these Fourier coefficients are

hkz pni D k Gpnduality D kz qnduality

zqnd D qnz

kz D1X

nD0qnz pn

which is an identity in Ha˝ It can be spelled out as

pnqnz 2 ˝ (337)

but then one has to be careful to notice that it only means that the differencebetween the right and left members has Ha˝-norm zero It is completely safeto insert (337) into (212) This gives for zw 2 ˝e

1 Ezw D h 1

qnzqnw (338)

So far we have not assumed anything about fpng besides it being an orthonormalbasis Later on we shall work with orthonormal polynomials (pn a polynomial ofdegree n) but if we for now just assume that p0 is a constant function more precisely

j˝j 1

then we find that the first dual basis vector is essentially the Cauchy transform

C˝z D h 1

z 1i D

q0z (339)

One may think of the two sides of (337) as representing the identity operator inHa˝ in two different ways with respect to the duality (334) The left member

is the Cauchy kernel which reproduces functions in Ha˝ by the Cauchy integralwhile the right member represents the identity as

Ppn ˝ qn where fpng is a basis

and fqng the corresponding dual basis (compare the right member in (331))Another representation of the identity in terms of the inner product in the Hilbert

space itself isP

pn ˝ Npn In the pointwise picture this spells out to

pnpnz z 2 ˝ (340)

However here there is no pointwise convergence If Ha˝ were a Hilbert spacewhich had continuous point evaluations the sum would converge to the reproducingkernel for the space but as is clear from previous discussions there are rarely anycontinuous point evaluations in Ha˝ This can be further confirmed by examplesIn Sect 72 the elements of Ha˝ are explicitly characterized in terms of powerseries in the case that ˝ is an annulus And many of these series turn out to be justformal power series having no convergence region at all This applies in particularto (340) (no convergence region)

310 Quadrature Domains

We single out in the present section a family of domains which are exceptionalfor Ha˝ This is the class of (finite) quadrature domains also called finitelydetermined domains [79] or algebraic domains [107] A few general references are[1 46 89 101] here we give just a short summary of some basic properties Theoriginal definition used in [1] says that a bounded domain ˝ C is a quadraturedomain if there exist finitely many points ak 2 ˝ and coefficients ckj 2 C such that

h dA DmX

nk1XjD0

ckjhjak (341)

for every function h which is analytic and integrable in ˝ If ˝ is just a boundedopen set which satisfies the same requirements we will still call it a quadraturedomain

Below is a list of equivalent requirements for a domain or open set to be aquadrature domain Strictly speaking the last two items iiindashiv are like theexponential transform itself insensitive for changes of ˝ by nullsets but onemay achieve equivalence in the pointwise sense by requiring that the domain ˝considered is complete with respect to area measure

i The exterior Cauchy transform is a rational function ie there exists a rationalfunction Rz such that

C˝z D Rz for all z 2 C n˝ (342)

310 Quadrature Domains 41

ii There exists a meromorphic function Sz in ˝ extending continuously to ˝with

Sz D Nz for z 2 ˝ (343)

This function Sz will be the Schwarz function (220) of ˝ [16 101]iii The exponential transform E˝zw is for zw large a rational function of the

E˝zw D Qz NwPzPw

where P and Q are polynomials in one and two variables respectivelyiv For some positive integer d there holds

detBkj0kjd D 0

Basic references for iiindashiv are [79ndash81] When the above conditions hold thenthe minimum possible number d in iv) the degree of P in iii and the numberof poles (counting multiplicities) of Sz in ˝ all coincide with the order of thequadrature domain ie the number d D Pm

kD1 nk in (341) For Q see moreprecisely below

If ˝ is connected and simply connected the above conditions indashiv are alsoequivalent to that any conformal map f W D ˝ is a rational function Thisstatement can be generalized to multiply connected domains in various ways seeeg [32 112]

Quadrature domains play a special role for the space H ˝ eg they make theanalytic part Ha˝ to collapse almost completely For non-quadrature domains theanalytic functions are dense in H ˝ but they still make up a rather brittle part ofthe space and there are usually no continuous point evaluations for these analyticfunctions (an example is given in Sect 72) The following theorem makes some ofthe above statements precise

Theorem 31 If ˝ is a not quadrature domain then the restriction of the map ˛ toO˝

˛jO˝ W O˝ H ˝

is injective and has dense range In particular Ha˝ D H ˝If on the other hand ˝ is a quadrature domain then ˛jO˝ is neither injective

nor has dense range Indeed the range is finite dimensional

dimHa˝ D d

where d is the order of the quadrature domain

Proof For the first statement we shall prove that˝ is a quadrature domain whenever˛jO˝ fails either to be injective or to have dense range

So assume that ˛jO˝ is not injective This means that h f f i D 0 for some

f 2 O˝ n f0g Multiplication by any rational function with poles outside ˝ is acontinuous operator in the H ˝-norm (see more precisely beginning of Sect 33)hence it follows that for any fixed w hellip ˝ the function 1

zw f z is zero as an elementof H ˝ Using (39) this gives that the exterior Cauchy transform of f vanishes

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

Invoking now Lemma 25 where in the formula (232) the exterior Cauchytransform of f is denoted h it follows that there exists a holomorphic function gin ˝ such that

Nzf zC gz D 0 z 2 ˝

Thus Sz D gzf z is a meromorphic function in ˝ satisfying (343) Hence ˝ is a

quadrature domain whenever ˛jO˝ is not injectiveNext assume that the range of ˛jO˝ is not dense in H ˝ By the Hahn-Banach

theorem the representation (38) of functionals and the definition (37) of H thismeans that there exists an analytic function f D Hg in ˝ which is not identicallyzero but which annihilates the analytic functions in the sense that

f zhzdAz D 0 for all h 2 O˝

Choosing hz D 1zw w hellip ˝ this gives the same relation as above hence we

conclude again that ˝ is a quadrature domainWhen ˝ is a quadrature domain the inner product evaluated on O˝ takes the

h f giHa˝ DX

Hak ajck Ncjf akgaj (345)

by (341) Here we have for notational convenience assumed that in (341) all themultiplicities nk equal one Thus the inner product only involves the values of thefunctions at d points and it follows that ˛O˝ has finite dimension equal to dand it then also coincides with its closure Ha˝

311 Analytic Functionals 43

311 Analytic Functionals

More general notions of quadrature domains may be discussed in terms of analyticfunctionals see [5 60 61] for this concept in general From a different point of viewthe term hyperfunction is also used thinking then on the concept as a generalizationof that of a distribution An analytic functional in an open set D C is simply alinear continuous functional on OD when this space is provided with the topologyof uniform convergence on compact sets Thus we denote the space of analyticfunctionals in D by O 0D A compact subset K D is a carrier for 2 O 0D iffor any open set K D an estimate

jhj c sup

jhj h 2 OD (346)

holds By definition of an analytic functional such a compact carrier always existsOn choosing h D kz D z1 one gets the natural definition of Cauchy

transform of an analytic functional 2 O 0D namely

Cz D 1kz z 2 Dc

Similarly one can define the double Cauchy transform the exponential transformetc One may also consider analytic functionals with compact carriers in ˝ aselements in H ˝ on defining the inner product by

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

This gives a version of the map ˛ in (36) going as

˛ W O 0˝ H ˝

It is enough to use test functions of the form h D kz in the definition of an analyticfunctional hence K is a carrier for 2 O 0D if and only if the Cauchy transformC has an analytic extension from C n D to C n K in the sense that there exists ananalytic function in CnK which agrees with C in CnD This makes it easy to handlecarriers For example one sees that the intersection between two convex carriers isagain a carrier This is because the complement of the union of the two convex setsis a connected neighborhood of infinity ensuring that the Cauchy transform extendsunambiguously to this set and then the Cauchy transform extends further to thecomplement of the intersection of the two convex sets by unique choice betweenthe two analytic continuation which exist by assumption As a consequence thereis always a unique minimal convex carrier See [61] Sect 47 for more details Arelated statement is that if there exists a carrier located on a straight line then thereis a carrier on this line which is the smallest of all carriers on the line see [60]Proposition 916

Now our main concern will be the analytic functional in the left memberof (341) or in view of (39) the functional represented by 1 2 H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

In this case K D ˝ is obviously a carrier with c D =Area˝ for any ˝ If ˝ is a quadrature domain with nodes a1 am 2 ˝ then also K D fa1 amgis a carrier but in general with a larger c Any carrier contains a carrier which isminimal with respect to set inclusion but there are usually many different minimalcarriers as the intersection between two carriers need not itself be a carrier In thecase of a quadrature domain as above K D fa1 amg is certainly a minimalcarrier but even in this case there are many other minimal carriers eg obtained bysweeping the mass sitting at the quadrature nodes to a surrounding curve

If the functional in (347) has a carrier K which is compactly contained in˝ (notonly in D) then ˝ is a ldquoquadrature domain in the wide senserdquo in the terminologyof Shapiro [101] In such a case there exists for any K a complex-valuedmeasure in such that

h dA DZ

h d h 2 OD (348)

One can think of as representing an element in O 0D or in H ˝ In the lattercase (348) says that

1 D as elements in H ˝

One cannot in general assert that the measure is real-valued (signed)when (348) holds and even less that it is positive unless one increases the supportconsiderably If in the case of a finite quadrature domain some of the nodes havecoefficients which are non-real then one need to connect them by curves in orderto support a real-valued representing distribution and then thicken the curve toget a real-valued measure which then still will be just a is signed measure Thereason one has to connect points is that complex coefficients mix the real andimaginary parts of an analytic test function and one need curves to resolve this seeExample 62 To get a positive representing measure (which amounts to being ableto choose c D =Area˝ in (346)) one may have to thicken the support quite alot But it is indeed always possible to find such a measure with compact supportin ˝ provided there is a compact carrier in ˝ at all see [26 48]

The search for minimal carriers which support positive measures leads to thenotion of potential theoretic skeleton or mother body to be discussed in Chap 6An illustrative example is the half-disk

˝ D fz 2 C W jzj lt 1 Re z gt 0g

By partial integration combined with analytic continuation the Schwarz functionsof the two boundary arcs one easily writes (348) on the specific form

h dA DZ

for any curve ˝ joining the points ˙i 2 ˝ Here D is a complex-valued measure on obtained from the jump ŒSz across between the twoSchwarz functions (precisely d D 1

i ŒSz dz along ) Thus any such curve isa carrier for M in fact a minimal carrier One can also choose to be a real-valueddistribution of order one on

So there are very many minimal carriers Taking the intersection of the convexhulls of them gives the unique minimal convex carrier namely the vertical segmentŒiCi This will however not be of much interest in connection with our mainproblem later on namely zeros of orthogonal polynomials What will be of moreinterest is that among all the above carriers there is a unique one which supportsa real-valued measure in fact a positive measure This is computed explicitly inSect 76 and found to agree quite well with our computational findings on locationsof zeros

Chapter 4Exponential Orthogonal Polynomials

Abstract This chapter contains some initial results obtained by direct or classicalmethods on the zeros of the exponential orthogonal polynomials For examplewe estimate the decay of coefficients of orthogonal expansion by using a result ofJL Walsh and we prove that the zeros always stay in the convex hull of the closureof the domain In addition we identify the zeros as also being eigenvalues of theappropriate Hessenberg matrices and we elaborate this connection further in thecase of quadrature domains

41 Orthogonal Expansions

If˝ is not a quadrature domain then the monomials f1 z z2 g are linearly inde-pendent in Ha˝ We shall consider the result of orthogonalizing this sequenceThe nth monic orthogonal polynomial which also can be characterized as the nthmonic polynomial of minimum norm will be denoted

Pnz D zn C terms of lower degree D ˘ njD1z zj (41)

z1 zn being the zeros of Pn Strictly speaking one should write zn1 znn The

corresponding normalized polynomial is

pnz D nzn C terms of lower degree n gt 0 (42)

The counting measure is

13n D 1

ızj (43)

We shall also use the notation

Vn D VPn D fzn1 znn g (44)

for this spectrum It is easy to see using (39) that z11 always is the center of massof ˝

48 4 Exponential Orthogonal Polynomials

As will become clear in Chap 5 the zero set Vn can also be characterized as thespectrum of the shift operator Z when this is truncated to the subspace of Ha˝

generated by f1Z1Z21 Zn11g It is also the spectrum of certain a Hessenbergmatrix see Sect 43 below

If ˝ is a quadrature domain of order d then 1 z zd1 are still linearlyindependent and can be orthonormalized Thus P0 Pd1 exist and are uniquelydetermined One can also define Pd namely as the unique monic polynomial ofdegree d with jjPdjj D 0 Thus Pd D 0 as an element of Ha˝ It is easy to seethat the zeros of Pd are exactly the quadrature nodes multiplicities counted

As is clear from the above if˝ is a quadrature domain then CŒz is always densein Ha˝ independent of the topology of ˝ However if ˝ is not a quadraturedomain then CŒz is dense in Ha˝ only under some topological assumptionSince the norm of Ha˝ is weaker than most standard norms on analytic functionsit is enough to assume that ˝ is simply connected in order to ensure denseness ofthe polynomials On the other hand polynomials are not dense if ˝ is an annulussee Example 72

For most of this chapter we assume that ˝ is simply connected Then CŒz isdense in Ha˝ so the polynomials pn (see (42)) make up an orthonormal basis inHa˝ (even in H ˝ if ˝ is not a quadrature domain) and any f 2 Ha˝ canbe expanded

nD0cnpn

with coefficients given by

cn D h f pni

where h f pni D h f pniH ˝ Of course we haveP jckj2 D jj f jj2 lt 1 but if f is

analytic in a larger domain there are better estimates of the coefficientsLet

g˝ez1 D log jzj C harmonic z 2 ˝e

be Greenrsquos function of the exterior domain (or open set) with a pole at infinityand having boundary values zero on ˝ Assuming that f is analytic in someneighborhood of ˝ define R D R f gt 0 to be the largest number such that f isanalytic in Cnfz 2 ˝e W g˝ez1 gt log Rg ie such that f is analytic in the unionof ˝ with the bounded components of C n˝ and in the unbounded component ofC n˝ up to the level line g˝ez1 D log R of g˝e1 Then we have

Proposition 41 With notations and assumptions as above

lim supn1

jh f pnij1=n 1

41 Orthogonal Expansions 49

Proof The theorem is an immediate consequence of a result by Walsh [108] onuniform approximation with f as in the theorem there exist polynomials Qn (bestuniform approximants) of degree n such that

lim supn1

jj f QnjjL1˝1=n 1

The rest is easy in view of the estimate (35) Since h f pni D h f Pn1kD0 ckpk pni

we have with Qn as above

jh f pnij jj f n1XkD0

ckpkjjH ˝ jjf Qn1jjH ˝ Cjj f Qn1jjL1˝

This gives the assertion of the propositionRecall from (337) the expansion of the Cauchy kernel

kz D 1

qnzpn (45)

where the coefficients

qnz D h 1

z pni (46)

make up the dual basis with respect to the Silva-Koumlthe-Grothendieck pairing Nowthe pn are orthonormal polynomials but this does not mean that the qnz arepolynomials In fact they cannot be since they tend to zero as z 1 but if ˝ is aquadrature domain they are at least rational functions

Expanding (46) near infinity in z gives since pn is orthogonal to allpolynomials of degree lt n

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

As a side remark from

zD pn pnz

one gets the somewhat remarkable identity

pnzqnz D h 1

zpn pni

which makes sense at least for z 2 ˝e

As an application of Proposition 41 we have since kz is analytic (as a functionof ) in all C n fzg that the radius of analyticity R f for f D kz with z in theunbounded component of ˝e is

Rkz D expŒg˝ez1

If˝ is a quadrature domain of order d then the dual basis qn in (46) can be madefairly explicit Indeed using (344) and (47) one concludes that the sum in (338) isfinite

1 Ezw Dd1XnD0

qnzqnw

and more precisely is of the form

1 Ezw Dd1XkD0

Pw (48)

where Qk is a polynomial of degree k (exactly) and Pz D Pdz is the monicpolynomial with zeros in the quadrature nodes as in (344) Cf [37] where thesame expansion was derived in a slightly different manner

In summary the dual basis is in the case of a quadrature domain given by qn D 0

for n d and

qnz D Qdn1zPz

for 0 n lt d

42 Zeros of Orthogonal Polynomials

The most direct way of characterizing the set of zeros of the monic orthogonal poly-nomials Pn is perhaps via the minimization problem that it solves Parametrizing ageneral monic polynomial of degree n by its zeros a1 an 2 C and setting

Ina1 an D jjnY

kD1z akjj2

kD1z ak

nYjD1 Nw Naj dAzdAw (49)

42 Zeros of Orthogonal Polynomials 51

we arrive at the problem

mina1an2C Ina1 an (410)

for which the unique solution is aj D znj (j D 1 n) up to a permutation Thevariational formulation of this problem says exactly that the minimizer Pnz DQn

kD1z zk is orthogonal to all polynomials of lower degreeOne readily checks that Ina1 an is a plurisubharmonic function of the

variables a1 an In fact computing derivatives gives that

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

that is the complex Hessian matrix is positive semidefiniteThe above statements are true for general Hilbert space norms but what is special

for the space Ha˝ is that the integral (49) in the definition of Ina1 an canmany times be pushed down to a smaller set For example whenever ˝ is smoothreal analytic it can be pushed down to a compact subset of˝ because the functionalh 7 hh 1i D 1

R˝ h dA h 2 OD in (347) will then have a carrier which is

compact in ˝ and when ˝ is a quadrature domain the integral can be representeddirectly on the quadrature nodes

Unfortunately it seems still difficult to push these facts to any specific conclusionby using directly the minimization of (49) For example it is evident from theexample to be given in Sect 76 that the zeros will generally not go into the uniqueminimal convex carrier of the analytic functional (347) Compare discussions at theend of Sect 311 What we can prove is only that they go into the convex hull of thedomain itself

Theorem 41 If ˝ is not a quadrature domain then

Vn conv ˝ (411)

for all n D 1 2 If ˝ is a quadrature domain of order d then (411) holds for1 n d

Proof Let D be a half plane containing˝ We need to show that Pn has no zeros inDe So to derive a contradiction assume that b 2 De is a zero of Pn and let a 2 Dbe the reflected point with respect to D Then za

zb Pnz is a monic polynomial ofdegree n different from Pnz even as an element in Ha˝ Thus

jjPnzjj lt jj z a

z bPnzjj

On the other hand when iii of Lemma 24 is expressed in terms of the H ˝

norm it shows that

jjPnzjj jj z a

z bPnzjj

This contradiction proves the theoremFurther results on the zeros of the orthogonal polynomials obtained by operator

theoretic methods will be given in Chap 5 In particular Theorem 41 will beproved again at the end of Sect 51

43 The Hessenberg Matrices

The zeros z1 zn can be characterized as being the eigenvalues of variousmatrices associated to ˝ for example matrices for truncated shift operators whichwe here discuss briefly from the point of view of Hessenberg matrices in acohyponormal setting Further studies related to hyponormal operator theory willfollow in Chap 5

We start from (42) and use again Z for the operator of multiplication by z inHa˝ Then

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

bkjpjzC bknpnz 0 k n 1

where the coefficients bkj D hZpk pji vanish whenever k C 1 lt j because pj isorthogonal to all polynomials of degree lt j Letting k run from zero to n 1 for a

43 The Hessenberg Matrices 53

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

bn10 bn11 bn12 bn13 bn1n1 bn1n

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

bn10 bn11 bn12 bn13 bn1n1

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

The n n matrix in the last expression is a lower Hessenberg matrix Wenotice that each zero of pnz is an eigenvalue of it the corresponding eigenvectorbeing the column vector with entries p0z pn1z evaluated at that zero Thetranspose matrix which is an upper Hessenberg matrix has the same eigenvaluesand when extended to a full semi-infinite matrix call it M it becomes that matrixwhich represents the operator Z with respect to the basis fpng1

nD0 in Ha˝ Thecommutation relation

ŒMM D 1 ˝ 1 D

then imposes stringent quadratic restrictions on the entries of MAs a preview of the coming examples and theoretical considerations we notice

that a Jacobi-Toeplitz matrix (ie three constant diagonals clustered around themain diagonal) satisfies the above commutator requirement Specifically take a b carbitrary complex numbers and define

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

or in a close form M D bS C aI C cS We denote by S the unilateral shift thatis the matrix with 1 under the main diagonal and zero elsewhere We know thatŒSS D e0 ˝ e0 hence the self-commutator of M becomes

ŒMM D jcj2 jbj2e0 ˝ e0 D

jcj2 jbj2 0 0 0 0 0

44 The Matrix Model of Quadrature Domains

The structure of the Hessenberg matrix touched in the last section is enhanced inthe case of a quadrature domain This topic was developed over two decades by theauthors and we only briefly comment below a few pertinent facts for the main bodyof these lecture notes Full details can be found in [40 80]

Let ˝ be a bounded quadrature domain of order d and let Ha˝ denote thed-dimensional analytic subspace of H ˝ The position operator Z leaves Ha˝

invariant and it is co-hyponormal on the larger Hilbert space H ˝ We considerthe scale of finite dimensional subspaces

Kn D spanfZkHa˝ W 0 k ng D spanfZkZm1 W 0 k n m 0gIn view of the commutation relation

ŒZZ D 1 ˝ 1

the operator Z leaves every subspace Kn invariantWe decompose the full space H ˝ into the hilbertian orthogonal sum

H ˝ D Ha˝˚ K1 K0˚ K2 K1˚ and accordingly Z inherits a block-matrix decomposition

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

The self-commutator identity yields

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

44 The Matrix Model of Quadrature Domains 55

AkZkC1 D Z

with the initial condition

ŒZ0Z0 C A0A

0 D 1 ˝ 1

The invariance of the principal function to finite rank perturbations of Z impliesthat

dimKnC1 Kn D dimHa˝

ker An D 0

for all n 0 See [80] for detailsWe have the luxury to change the bases in the wandering spaces and identify them

all with the same Hilbert space of dimension d Moreover in this identification wecan assume that all transforms An n 0 are positive Consequently we obtain thesimilarity relations

ZkC1 D AkZkA1k k 0

The factorization of the exponential transform attached to these data yields forlarge values of jzj jwj

1 E˝zw D hZ z11 Z w11i D hZ0 z11 Z0 w11i

Therefore the spectrum of the matrix Z0 coincides with the quadrature nodesmultiplicity included We read from here that the domain ˝ coincides modulofinitely many points to the super level set of the localized resolvent of Z0

˝ D fz 2 C W kZ0 z11k gt 1g

Having the formula for the Cauchy transform of the uniform mass on ˝ embeddedinto the exponential transform factorization also gives a formula for the Schwarzfunction

Sz D z hZ0 z11 1i C hZ z11 1i

In general when regarded as a rational function the localized resolvent of amatrix appearing in the above computations carries a special algebraic structure

Proposition 42 Let A be a complex d d matrix with cyclic vector 2 Cd Then

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

where Pz is the minimal polynomial (of degree d) of A and Qkz are complexpolynomials subject to the exact degree condition

deg Qk D k 0 k d 1

The converse also holds true For proofs and a discussion of the geometricimplications of such a hermitian square decomposition see [37] An algebraic curveof equation

jPzj2 Dd1XkD1

jQkzj2

is known as a generalized lemniscate and it is suitable for studying the globalSchwarz reflection into it

In the setting of the functional model with ˝ is a quadrature domain of order dA D Z and D 1 then Ha˝ can be identified with C

d and (412) becomes thesame as (48)

Chapter 5Finite Central Truncations of Linear Operators

Abstract By interpreting the exponential orthogonal polynomials as characteristicpolynomials of finite central truncations of the underlying hyponormal operator oneopens a vast toolbox of Hilbert space geometry methods In particular we prove inthis chapter that trace class modifications of the hyponormal operator attached to adomain will not alter the convex hull of the support of any cluster point of the countin measures of the roots of the orthogonal polynomials As a sharp departure fromthe case of complex orthogonal polynomials associated to a Lebesgue space weprove that the convex hull of these supports is not affected by taking the union of anopen set with a disjoint quadrature domain However similar to the case of Bergmanorthogonal polynomials we prove that the exponential orthogonal polynomialssatisfy a three term relation only in the case of an ellipse Some general perturbationtheory arguments are collected in the last section

51 Trace Class Perturbations

A Hilbert space interpretation of the orthogonal polynomials considered in thepreceding sections touches the classical topics of the asymptotics of the spectra offinite central truncations of a linear bounded operator along the Krylov subspacesassociated to a privileged (cyclic) vector

We start by recalling (with proofs) some well-known facts Let A 2 L H be alinear bounded operator acting on the complex Hilbert space H and let 2 H be anon-zero vector We denote HnA the linear span of the vectors A An1and let n be the orthogonal projection of H onto this finite dimensional subspaceOf interest for our study is the asymptotics of the counting measures n ofthe spectra of the finite central truncations An D nAn or equivalently theasymptotics of the evaluations of these measures on complex analytic polynomials

Zpzdnz D tr pAn

n p 2 CŒz

58 5 Finite Central Truncations of Linear Operators

Indeed the monic orthogonal polynomials Pn in question minimize the func-tional (semi-norm)

kqkA D kqAk q 2 CŒz

and the zeros of Pn (whenever they are unambiguously defined) coincide with thespectrum of An To be more specific the following well-known lemma holds

Lemma 51 Assume that HnA curren HnC1A and let the polynomial Pnz Dzn C qn1z deg qn1 n 1 satisfy

PnA HnA

Then detz An D Pnz

Proof Remark that for every k n 1 we have

Akn D nAnAn nAn D nAk

By the assumption HnA curren HnC1A the vectors An An1n are

linearly independent and they generate the subspace HnA that is is a cyclicvector for An According to the Cayley-Hamilton theorem the minimal polynomialof An coincides with its characteristic polynomial Qnz In particular

QnAn Akn k lt n

One step further for any k lt n one finds

hQnAAki D hQnA nAki D hQnAnAki D 0

Thus Qn D PnAn immediate consequence of the above observation is that the spectra of the

finite rank central truncations An are contained in the closure of the numerical rangeWA of A Indeed

An WAn WA

We recall that the numerical range of A is the set

WA D fhAx xi W x 2 H kxk D 1g

A basic theorem of Hausdorff and Toeplitz asserts that WA is a convex set whichcontains the spectrum of A in its closure

Next we prove that the asymptotics of the counting measures of the truncationsAn is not affected by trace-class perturbations Denote by jAj1 D tr

pAA the trace

norm of an operator A and by C1H the set of those A 2 H with jAj1 lt 1

51 Trace Class Perturbations 59

Proposition 51 Let AB 2 L H with AB of finite trace AB 2 C1H Thenfor every polynomial p 2 CŒz we have

tr pAn tr pBn

Proof It suffices to prove the statement for a monomial pz D zk Denote C DA B the trace class difference between the two operators Using then the generalidentity

Akn Bk

jD1Aj1

n An BnBkjn

it follows that there exists a polynomial Sku v with positive coefficients with theproperty

jtrAkn Bk

nj SkkAnk kBnkjAn Bnj1

Since jAn Bnj1 jCj1 one finds

jtrAkn Bk

nj SkkAk kBkjCj1and the proof is complete

Corollary 51 Let n 13n denote the counting measures of the spectra of Anrespectively Bn above Then

limn1Œ

uniformly on compact subsets which are disjoint of the convex hull of A[ BSimilarities which are implemented by almost unitary transformations (in the

trace-class sense) also leave invariant the asymptotics of our counting measures

Corollary 52 Let U 2 L H be a unitary operator and L 2 C1H a trace-class operator so that X D U C L is invertible Let A 2 L H be a linearbounded operator with a distinguished non-zero vector 2 H Denote B D X1AX D X1 and consider the finite central truncations An Bn along the subspacesHnA respectively HnB Then for every complex polynomial p 2 CŒz wehave

tr pAn tr pBn

We put the above techniques at work and elaborate another proof of Theorem 41This goes as follows Let T 2 L H be a hyponormal operator and 2 H

a distinguished vector The monic orthogonal polynomial with respect to theseminorm kpTk is denoted as above by Pnz Then the claim is that all zerosof Pn lie in the convex hull of the spectrum of T

Indeed assume that one zero z D does not belong to the convex hull of thespectrum of T Let ` denote a real line separating strictly from the convex setWT Denote by the symmetric point of with respect to this line Then theoperator T is invertible hyponormal and its norm coincides with the spectralradius Hence

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

which contradicts the minimality of kPnTk

52 Padeacute Approximation Scheme

The specific positivity structure of the exponential transform of a bounded planardomain imposes an adapted Padeacute approximation scheme This rational approxima-tion becomes exact in the case of quadrature domains and it is useful in treating allsemi-normal operators which share the same principal function

We work with an irreducible hyponormal operator of T 2 L H with rank-oneself-commutator

ŒTT D ˝

The associated characteristic function that is the exponential transform of aprincipal function g is

Ezw D detT zT wT z1T w1 D

D 1 hT w1 T z1i D 1 1X

zkC1w`C1

Fix a positive integer N and denote by TN the finite central truncation of theoperator T to the linear subspace generated by the vectors T TN1Then

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

52 Padeacute Approximation Scheme 61

Thus it is natural to consider the rational function

ENzw D 1 hTN w1 T

as an optimal approximate of Ezw in a neighborhood of the point at infinityRemark that

ENzw D QNzw

PNzPNw

where PN is the associated orthogonal polynomial whenever it is unambiguouslydefined and the polynomial kernel QNzw is positive semi-definite and has degreeat most N 1 in each variable

A minimal number of assumptions characterizes in fact the rational approxima-tion by EN as follows

Theorem 51 Let Ezw D 1P1k`D0

zkC1w`C1 be the exponential transform of ameasurable function of compact support g 0 g 1 attached to the hyponormaloperator T Fix a positive integer N

There exists a unique formal series

Ezw D 1 1X

zkC1w`C1

with the matching property

ck` D bk` for 0 k N 1 0 ` N or 0 k N 0 ` N 1

and positivity and rank constraints

ck`1k`D0 0 rankck`

1k`D0 minN n

where n D rankbk`Nk`D0

In this case Ezw D ENzw Moreover Ezw D Ezw as formal series ifand only if the function g is the characteristic function of a quadrature domain oforder d N

Proof Assume the infinite matrix ck`1k`D0 is subject to the two conditions in the

statement Then either detck`N1k`D0 D detbk`

N1k`D0 D 0 or detck`

N1k`D0 gt 0

In the first case we know that E is the exponential transform of a quadraturedomain of order d N 1 and there exists a unique positive semi-definiteextension rank preserving extension of the matrix bk`

N1k`D0 Then necessarily

Ezw D ENzw D EzwIn the second situation condition detck`

Nk`D0 D 0 defines unambiguously the

entry cNN Then again there is a unique infinite matrix completion of ck` which

preserves rank and semi-positivity In addition we identify

ck` D hT`N T

first for k ` N and then for all values of k `The difference between the exponential transform and its diagonal Padeacute approx-

imant above is easy to control outside the convex hull of the support of the originalfunction g

Corollary 53 Under the assumptions in the theorem above let K D conv suppgdenote the convex hull of the closed support of the function g and let F be a compactset disjoint of K Then

limN1 jENzw Ezwj D 0

uniformly for zw 2 F

Proof The closed support of the principal function g is equal to the spectrum ofthe irreducible hyponormal operator T Let c 2 C be an auxiliary point Since thespectral radius of T aI is equal to its norm there exists a center c and radius R gt 0so that K DcR and j cj gt R for all 2 F Let R0 D inf2F j cj andremark that R0 gt R as F is compact Then we can write Neumann series expansionsof the two kernels centered at c

Ezw D 1 hT c w c1 T c z c1i D

hT ck T c`iw ckC1z c`C1

According to the above theorem

Ezw ENzw D hT cN T cNiw cNC1z cNC1 hT

N cN TN cNiw cNC1z cNC1 C

1XkgtN or `gtN

hTN ck TN c`iw ckC1z c`C1

Remark that kTN ck kT ck In conclusion for all zw 2 F we obtain

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Finally we record below a simple observation pertaining to the matrix approx-imation of the multiplier Z D Mz in the functional space H ˝ Specifically

53 Three Term Relation for the Orthogonal Polynomials 63

passing to the final central truncations Zn we obtain

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

Compare (337) or (45) We recall that pk are the orthogonal polynomials associ-ated to ˝ while qkz D h 1

z pki make up the dual basis see (46)

53 Three Term Relation for the Orthogonal Polynomials

We show in analogy with the known case of Bergman orthogonal polynomialsthat the ellipse is the only bounded domain for which the exponential orthogonalpolynomials satisfy a three term relation

From the very beginning we exclude the case of a quadrature domain wherehigher degree exponential orthogonal polynomials cannot be defined withoutambiguity

Theorem 52 Let˝ be a bounded open set in C which is not a quadrature domainThe exponential orthogonal polynomials satisfy a three term relation if and only if˝ is an ellipse

Proof Let T 2 L H denote the irreducible hyponormal operator with rank-oneself commutator and spectrum equal to the closure of˝ Since˝ is not a quadraturedomain the space H is spanned by the orthonormal system pnT n 0 wherepn are the exponential orthonormal polynomials and ŒTT D ˝ A three termrelation for the orthogonal polynomials is equivalent to the matrix representation ofT by three non-zero diagonals Indeed the assumption is

zpnz D cnC1pnC1zC anpnzC bnpn1z

where an bn cn are complex numbers and p1 D 0 Hence

TpnT D cnC1pnC1T C anpnT C bnpn1T

The matrix representations of T and T are

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

The self-commutator is represented in the same basis as

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

where r gt 0Writing on the matrix elements the commutator equation yields a system of non-

linear equations

ja0j2 C jb1j2 D r C ja0j2 C jc1j2 jc1j2 C ja1j2 C jb2j2 D jb1j2 C ja1j2 C jc2j2

a0c1 C b1a1 D a0b1 C c1a1 a1c2 C b2a2 D a1b2 C c2a2

b1c2 D c1b2 b2c3 D c2b3

We infer from the first relations

jbkj2 D r C jckj2 k 1

in particular bk curren 0 k 1

If there exists cn D 0 then the last string of relations imply ck D 0 for all k 1But then T D that is the cyclic space of T with respect to the vector hasdimension one and this occurs only if ˝ is a disk This case was excluded from thebeginning

We can assume therefore that all matrix entries ck k 1 are non-zero Then

jbkj2jbkC1j2 D jckj2

jckC1j2 D r C jckj2r C jckC1j2 k 1

This implies

jb1j D jb2j D jb3j D jc1j D jc2j D jc3j D

Further on we can pass to a unitary equivalence UTU without changing thespectrum of T or the assumptions on the three term recurrence relations With U Ddiag0 1 2 we achieve

b1 D b2 D b3 D D s gt 0

Then the third string of relations imply

c1 D c2 D c3 D D u 2 C

Finally the second string of relations yield

uak C sakC1 D uakC1 C sak k 0

Consequently

uak sak D ua0 sa0 k 0

Since juj curren s these equations have unique solution

a1 D a2 D a3 D D a

The translation T 7 T aI does not change the statement so we can assumea D 0 In conclusion T has zero on the main diagonal and constants on the borderingsubsuper diagonals Denoting by S the unilateral shift on `2N we obtain T DuS C S That is the spectrum of T is an ellipse

54 Disjoint Unions of Domains

It is legitimate to ask what happens with the zeros of the exponential orthogonalpolynomials when two disjoint supporting sets are put together The case of Szegoumlor Bergman orthogonal polynomials was thoroughly studied by Widom [110]respectively Saff Stylianopoulos and the present authors [50]

Let Ai 2 L Hi i D 1 2 be two linear bounded operators and fix somenon-zero vectors i 2 Hi Assume that the spectra Ai i D 1 2 are disjointThe multiplicative property of the exponential transform leads to the followingconstruction suppose that there exists a linear operator A 2 L H with adistinguished vector 2 H so that

1 kA1 z11k21 kA2 z12k2 D 1 kA z1k2for jzj large enough The main question is what is the relation between theasymptotics of the spectra of the finite central truncations of A and those of A1A2along the subspaces HnA respectively HnAi i

To start investigating this question we polarize the identity above and rearrangethe terms

hA1 z11 A1 w11i C hA2 z12 A2 w12i D

hA z1 A w1iC

hA1 z11 ˝ A2 z12 A1 w11 ˝ A2 w12iRegarding the latter as the Gram matrix of a family of vectors we infer that there

exists an isometric transformation

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

The tensor product of two resolvents can be linearized by the usual resolventequation trick Note first that the elementary operators I ˝ A2 A1 ˝ I commuteand their spectra are disjoint Hence I ˝ A2 A1 ˝ I is invertible by the spectralmapping theorem for analytic functional calculus of commuting pairs Then

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

54 Disjoint Unions of Domains 67

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

By integrating the above identity along a large circle with the differential formpzdz we get for an arbitrary polynomial p 2 CŒz

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

We know that a trace-class perturbation will not alter the asymptotics ofthe orthogonal polynomials associated to these pairs of operators and vectorsTherefore assuming that dim H2 lt 1 we can work only with polynomials pzwhich annihilate A2 W pA2 D 0 We obtain then an isometric map

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

We introduce the operator D W H1 H1 ˝ H2

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

and remark that kDxk kxk for x D pA11 pA22 D 0 Note also that

A1 ˝ ID D DA1 (51)

With these preparations we are ready to prove the following surprising resultabout the invisibility of quadrature domains in the asymptotics of the zeros ofexponential orthogonal polynomials associated to disjoint unions of open sets

Theorem 53 Let ˝ D ˝1 [ ˝2 be a disjoint union of a bounded open set ˝1

with a quadrature domain˝2 The zeros of the exponential orthogonal polynomialsassociated to ˝ cluster in the convex hull of ˝1

Proof Modulo changing the complex variables into the conjugate ones we areworking in the setting described above A1A2 are the cohyponormal operatorsattached to ˝1 respectively˝2 A is attached to ˝ etc

Let Pn denote the orthogonal polynomials attached to ˝ that is orthogonalizingthe semi-norm kpAk Since a finite rank perturbation does not affect the convexhull of the zero asymptotics we can work only in the subspace of polynomialsp 2 CŒz which annihilate A2 along the vector 2 that is pA22 D 0 In particularfor these polynomials we have

kpA11k2 kDpA11k2 D kpAk2

or by polarization and using the intertwining relation (51)

hA1 cx xi hA1 c˝ IDxDxi D hAy yi

where c 2 C and x D pA11 y D pAAssume that r ReA1 c r for a non-negative r Then obviously ReA

c˝ I satisfies the same bounds and because

kxk2 kDxk2 D kyk2

we obtain

rkyk2 RehA cy yi rkyk2

This proves that any weak limit of the zeros of the orthogonal polynomialsPn has support contained in the numerical range of A1 hence the statement of thetheorem

Since the operator A1 ˝ I I ˝ A2 is cohyponormal invertible if dist˝1˝2 gt

0 and k2k2 D Area˝2 we can complement the above statement by thequantitative observation

Area˝2p dist˝1˝2

Numerical experiments illustrating the asymptotic zero distribution in the caseof an external disk and also comparison with the corresponding behavior forBergman orthogonal polynomials can be found in Sects 781 and 782 belowFigs 711ndash714 The zeros ignore the disk in sharp contrast with the situationfor a disjoint union of two non-quadrature domains illustrated by Fig 715 inSect 7101

55 Perturbations of Finite Truncations

Let ˝ be a bounded quadrature domain with mother body (this concept to beprecisely defined in Sect 61) a positive measure supported on a compact subsetK ˝ Specifically we here just assume just that the functional f 7 R

( f 2 O˝) has a compact carrier (in˝) which by discussions in Sect 311 meansthat there exists and K as above such that

f dA DZ

Kf d f 2 O˝

We assume that ˝ is not a finite quadrature domain

55 Perturbations of Finite Truncations 69

The inner product in the space H ˝ can in this case be pushed to the set K aswe know

h f gi D 1

ZHzwf zf wdzdw

As in previous sections we denote by the same letter the positive operator

Hf w D 1

ZHzwf zdz

We will be interested in evaluating this operator on polynomials f 2 CŒz remarkingthat H is Hilbert-Schmidt (and even more smooth) on L2 due to the compactnessof K and the analyticityanti-analyticity of the integral kernel Hzw

We will perform simultaneously spectral analysis approximation in two non-equivalent norms Besides the inner product h i of the space H ˝ we alsoconsider the inner product in the Lebesgue space L2 The norm in H ˝

will be simply denoted k k while the norm in L2 carries a subscript k k2The orthonormal polynomials in H ˝ are denoted as before by pn

pnz D nzn C Ozn1

while the orthonormal polynomials in L2 are

qnz D nzn C Ozn1

The significance of the leading coefficients n n is classical

1n D inf

deg f n1 kzn f k 1n D inf

deg f n1 kzn f k2

Finally to fix notation we denote by n the orthogonal projection of H ˝ ontothe polynomials of degree less than or equal to n 1 and by n the correspondingorthogonal projection in L2 For an linear operator T on one of the two Hilbertspaces we denote Tn D nTn or Tn D nTn the respective compressions Asproved before the operator Z D Mz of multiplication by the variable on H ˝

has finite central truncations Zn with characteristic polynomials equal to pn whilethe normal operator A D Mz on L2 produces finite central truncations An whosecharacteristic polynomials are the orthogonal polynomials qn

Let f g 2 Cn1Œz be polynomials of degree less than or equal to n 1 Then

hZf gi D h f Zgi D Hf zg D zHf g D AHf g

and one step further we can keep track of the orthogonal projections and insertwhenever possible the corresponding orthogonal projections

hZf gi D hnZn f gi Hf zg D nC1Hn f Ang etc

We end up with the identity

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

Since˝ is not a finite point quadrature domain the operator Hn is strictly positiveon Cn1Œz

We expect in general that the difference

HnZn H1

n An D nAnC1 nHnH1

converges to zero in the operator topology which would provide the expectedasymptotics for the spectrum of Zn To be more specific we isolate the followinggeneral observation We denote by conv K the convex hull of the compact set K

Proposition 52 Assume in the above notation that

lim sup kH1n ZnHn Ank D r lt 1

Let nk 2 Znk k 1 be a sequence of spectral points converging to Then

dist conv K r

Proof We drop the subsequence notation and consider a unit vector un with theproperty

H1n ZnHnun D nun

Since A is a normal operator the point n D Anun un belongs to the numericalrange of A that is conv K But

jn nj D jH1n ZnHnun Anun unj kH1

n ZnHn Ank

and the statement follows by passing to the limitNext we analyze in more detail the defect operator D

n D H1n ZnHn An The

difference of two orthogonal projections in its expression is rank one

nC1 n D qn qn

whence

Dn WD nAnC1 nHnH1n D nAqn qnHnH1

n D nAqnH1n nHqn

The good news is that we can further simplify this rank one matrixFirst remark that

Aqn1 D zn1zn1 C Ozn1 D n1n

qnzC Ozn1

and consequently

nAqn qn1 D qnAqn1 D n1n

nAqn qk D qn zqk D 0 k n 2

we infer

nAqn D n1n

The other factor in the difference operator Dn can be simplified as well For anarbitrary vector f one has

HnH1n f qn D f H1

n nHqn

We decompose in orthogonal components

Hqn D s C t deg s n 1 nt D 0

On the other hand there exists a polynomial h 2 Cn1Œx which satisfies

Hh D s C t0 nt0 D 0

By its definition s D Hnh hence

h D H1n nHqn

By subtracting the two equations we find that the polynomial qn h of degree equalto n satisfies Hqn h Cn1Œz in the inner product of L2 That is qn h isorthogonal to Cn1Œz in the space H ˝ Consequently qn h is a scalar multipleof pn We find

h D qn n

by Cramerrsquos rule for computing the inverse of a matrix

Putting all these computations together we arrive at the following statement

Theorem 54 Let ˝ be a bounded open set which admits a quadrature identity foranalytic functions with respect to a positive measure supported by a compact subsetK of ˝ If

npnk2 D 0

then any weak limit of the counting measures of the exponential orthogonalpolynomials is supported by the convex hull of K

The case of an ellipse is extreme for the above theorem because the systems oforthogonal polynomials pn and qn coincide up to the right normalization thatis qn D n

npn for all n 0

For regular measures the general theory of orthogonal polynomialsimplies that the quotient n1

nconverges to 1=capK hence only condition

limn1 kqn nn

pnk2 D 0 suffices for the spectral asymptotics

Corollary 54 Assume under the hypotheses of the theorem that the closed supportK of the quadrature measure decomposes into a disjoint union K D fg [ K0 Thenany weak limit of the counting measures of the exponential orthogonal polynomialsis supported by the convex hull of K0

Proof We know that a finite rank perturbation of the operator Z will not change theasymptotics of the exponential orthogonal polynomials Then we repeat the proofof the theorem by considering orthogonal projections onto the maximal ideal ofpolynomials vanishing at Remarking that the vanishing condition is still valid forthe new sequence of projections

The computations of the preceding section can be better understood from thepoint of view of Cholesky decomposition of positive definite matrices The notationis the same In the end we will need to work with the monic orthogonal polynomialsse we denote for all n 0

Pnz D pnz

nD zn C nzn1 C lower order terms

Qnz D qnz

nD zn C ınzn1 C lower order terms

We still have hPnPki D QnQk D 0 for all k curren nThe connection between orthogonal polynomials and moment matrices is well

known and can be derived from the decompositions

zn D rnnpnzC rnn1pn1zC

zn D snnqnzC snn1qn1zC

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

Denoting by B D hzn zki1nkD0 and N D zn zk1nkD0 the corresponding momentmatrices and by R D rj`

1j`D0 S D sj`

1j`D0 the lower triangular matrices above

we obtain Cholesky decompositions

B D RR N D SS

Note that the rows of the inverse matrices R1 and S1 coincide with thecoefficients of the orthogonal polynomials pn respectively qn as derived from theidentities

1 z z2 z3 T D R p0z p1z p2z T

1 z z2 z3 T D Sq0z q1z 22z T

The transition matrix C entering into the decomposition

pn DXkn

is therefore

C D R1S

Remark that C1 is Hilbert-Schmidt because

ınm D Hpn pm DXk`

cnkHqk q`cm`

or in closed matricial form

I D CHC

The quantitative defect in the spectral asymptotic theorem above is

kqn pn

cnnk22 D

n1XkD0

And we want in the best scenario that this sequence converges to zero One sufficientcondition is stated below

Proposition 53 Assume that the measure is regular and that the matrix Hassociated to the H-kernel with respect to the orthonormal basis of polynomialsin L2 admits the LDU decomposition

H D I C LDI C L

where D is a diagonal operator and L is a Hilbert-Schmidt strictly lower triangularoperator Then any weak limit of the counting measures of the exponentialorthogonal polynomials is supported by the convex hull of K D supp

Again the ellipse is relevant as in this case H D D

56 Real Central Truncations

There is flexibility in chopping out the quantized hyponormal operator T associatedto a bounded open set ˝ of the complex plane Leaving aside the Krylov spacesattached to T and the range of self-commutator we can envisage a ldquorealrdquo cut-offrather than the ldquocomplexrdquo truncations used throughout these notes

Quite specifically let ŒTT D ˝ be the irreducible hyponormal operatorwith principal function gT D ˝ We consider the linear subspaces

Vn D spanfTiTj maxi j ng

and the orthogonal projections n onto them Let Rn D nTn be the compression ofT to the finite dimensional subspace Vn Clearly n monotonically converges in thestrong operator topology to the identity Henceforth we call Rn the n-th order realcentral truncation of T

Note that due to the commutation relation ŒTT D ˝ we have

TVn VnC1 TVn VnC1

That is with respect to the chain Vn the matrix attached to the operator T is blockthree-diagonal

The quantized matrix model constructed for a quadrature domain cf Sect 44yields the following conclusion

Theorem 55 Let ˝ be a quadrature domain of order d with nodes a1 am andrespective multiplicities d1 dm The counting measures n of the spectra of thereal central truncations of the hyponormal operator attached to ˝ satisfy

w limnn D d1

dıa1 C d2

dıa2 C C dm

56 Real Central Truncations 75

Proof The staircase matrix model of T proved in Sect 44 (for the adjoint transposeZ) shows that

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

Recall that the diagonal blocks Tk are all similar and they have spectrum equalto fa1 amg with the same multiplicities fd1 dmg as the original quadraturedomain

We also know that the vectors T Td1 are linearly independent butTd is linear dependent of them For any integer n gt d we have using the notationof Sect 44

Vn D spanfTiTj maxi j ng D spanfTiHa˝ i ng D Kn

In other terms the real truncation Rn consists of the first d C 1 blocks of thestair-case matrix model of T Remark also that

dim Vn D n C 1d

Let pz be a complex polynomial The matrix pTn has a lower triangular block-structure with pTk 0 k n as diagonal blocks Therefore

tr pRn DnX

kD0tr pTk D n C 1pT0 D n C 1Œd1pa1C C dmpam

The normalized traces give exactly the value of the counting measure

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

In conclusion we face a constant sequence of counting measures convergent tothe finite atomic measure in the statement The convergence established on complexpolynomials obviously extends to all real polynomials

It would be interesting to explore the following question Let ˝ be a domainsatisfying a generalized quadrature identity with positive measure 13 By approxi-mating 13 by a sequence of finite atomic positive measures 13n we get a sequence ofquadrature domains ˝ converging ldquoin momentsrdquo to ˝ The attached operators Trespectively Tn also converge in moments Can we compare the real truncations ofT to the real truncations of Tn

Chapter 6Mother Bodies

Abstract We outline the general theory for a certain kind of potential theoreticskeletons or lsquomother bodiesrsquo associated to a given domain The hope is generallyspeaking that such skeletons can be identified as attractors for zeros of orthogonalpolynomials and in a few cases such expectations have indeed been met theoret-ically andor experimentally For the exponential polynomials the success is ratherlimited so far but by building in enough flexibility in the models one expects in somenot so distant future to reach a reasonable matching In the present chapter we set updesirable properties (formulated as lsquoaxiomsrsquo) to be satisfied by mother bodies Sincethe search for potential theoretic skeletons is a highly ill-posed problem (related tothe Cauchy problem for an elliptic operator) very few domains admit mother bodiesbut for domains with piecewise algebraic boundaries there is a rather constructiveand efficient theory bearing in mind that the same class of domains is also amenablefor studying zeros of orthogonal polynomials

61 General

We have previously discussed the notion of carrier for the analytic functional f 7h f 1iH ˝ D 1

R˝ f dA f 2 O˝ see Sect 311 Of special interest are minimal

carriers which always exist and it is natural to conjecture that they have somethingto do with asymptotic locations of zeros of orthogonal polynomials

One thing which speaks against minimal carriers is that there may be too manyof them as was seen in an example in Sect 311 Therefore a more refined notion isdesirable Such a refinement may be based on turning from analytic test functionsto harmonic and subharmonic functions hence from complex analysis to potentialtheory In this area there is a more or less established notion of potential theoreticskeleton or ldquomother bodyrdquo to use a term coined by the Bulgarian geophysicistZidarov [113] and the group around him

Mother bodies have the advantage of not being too abundant (there may befinitely many of them for a given body but probably not continuous families) Theprice to be payed for this is that they do not always exist Existence is in factextremely rare in a general setting However for the types of domains discussedin this paper like domains with piecewise algebraic boundaries finitely manygood candidates can always be singled out and then it becomes a task of partly

78 6 Mother Bodies

combinatorial nature to figure out which of these really are mother bodies or atleast are good enough for the purpose at hand In contrast to a carrier which is a seta mother body is a mass distribution (a measure) However it will turn out that onlythe support of this (again a set) will have a reasonable agreement with zero sets oforthogonal polynomials This will be exemplified in Chap 7

Informally a mother body for a given body (mass distribution) is a moreconcentrated body (a ldquoskeletonrdquo) which produces the same exterior gravitationalfield (or potential) as the given body The basic example which actually goesback to Isaac Newton (in the case of three dimensions) is the appropriate pointmass at the center of a massive homogenous ball Much of our inspiration forconsidering mother bodies comes directly from D Zidarov and members in hisgroup for example O Kounchev But also specific questions by HS Shapiro andconsiderations in general around quadrature domains have been formative (see[101])

First some notational issues We define the logarithmic potential of a measure as

Uz D 1

Zlog jz j d

so that U D Some variants of this notation will also be used for example ifthe measure has a density with respect to area measure (ie d D dA) we writeU and U˝ if D ˝ We shall also use

Cz D 1

for the Cauchy transform of a measure so that Nz C D

The given body will be represented for us by a positive density 0 on abounded open set ˝ C extended by D 0 outside ˝ Thus the body is anabsolutely continuous measure with compact support represented by its densityfunction 2 L1C Our assumptions on ˝ will always include that ˝ equalsthe interior of its closure and that ˝ has area measure zero We may write theseassumptions as

˝e D ˝ j˝j D 0 (61)

Quite often we will assume more The case that will be mostly discussed is that ofa uniform density say D ˝ Ideal requirements for a mother body of werediscussed in [34] where the following list of desirable properties was singled out

Definition 61 A mother body for a given mass distribution 0 as above is ameasure satisfying the following requirements

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

Every component of C n supp intersects ˝e

The positivity of has been stated as a separate axiom M3 because in somecontexts it turns out to be appropriate to keep open for signed measures if nogood positive measures can be found However further relaxations for exampleto distributions of higher order turns out to be no good because it allows for toomany mother bodies

It is obvious from the axioms that supp ˝ and in general supp will reachout to ˝ (at some points) However does not load ˝ ie ˝ D 0 Thiscan be deduced from a generalized version of Katorsquos inequality see [12 28] Indeedsetting u D UU so that u 0 by M2 this inequality shows that the restrictionof the measure u to fu D 0g is non-negative Under our assumptions this impliesthat D 0 on ˝ It is worth to notice also that u is lower semicontinuous (becauseU is) and that therefore u D 0 automatically holds on all ˝c (not only ˝e aswritten in M1)

The two inequality statements above M2 and M3 concern orderings whichare dual to each other with respect the pairing between potentials and measuresappearing in the definition of mutual energy

13energy DZ

Ud13 DZ

So the partial order in M2 is a natural ingredient in potential theory and in fact thisaxiom is necessary in order that one shall be able to reconstruct from by meansof the appropriate balayage process partial balayage For this process a desireddensity 0 has to be given in advance in all C Then one wants to sweep

80 6 Mother Bodies

satisfying M3 M4 to a body (measure) of the form (in terms of densities)

for some open set˝ C (not known in advance) That is always possible providedthere is enough space under Specifically it is enough to assume that

Rd ltR

dA see [41] but to simplify the treatment it is convenient to impose bounds0 lt c1 c2 lt 1 The open set ˝ is in any case uniquely determined (up tonull-sets) by the requirement that M1 M2 shall hold

We shall not go deeply into this kind of potential theory but for later use weintroduce at least the notation

Bal D ˝ (62)

for the partial balayage 7 D ˝ We have assumed that satisfies M3and M4 In a different language (62) says that ˝ is a quadrature domain forsubharmonic functions with respect to (and the given density )

dA 2 SL1˝ (63)

This formulation requires some more care as for nullsets but we shall not go intosuch details In general we refer to [89] SL1˝ denotes the set of integrablesubharmonic functions in ˝

Like mother bodies also partial balayage can be traced back to Zidarov [113]at least as for discrete and numerical versions From a mathematical point ofview it has been (independently) developed by Sakai [89] and one of the presentauthors [35 42] It is closely related to weighted equilibrium distributions [86]obstacle problems [90 91] the ldquosmash sumrdquo [17] and stochastic processes suchas internal diffusion limited aggregation [71] as well as several growth processesin mathematical physics and corresponding issues in random matrix theory [57]Further examples of recent developments are [27 41]

Now the point we wish to make (eventually) in this work is that we want to matchasymptotic distribution of zeros of orthogonal polynomials ie weak limits of thecounting measures 13n with mother bodies for suitable choices (yet unclear) of In the case of Bergman polynomials there has been some progress in this respectIn that case the appropriate given body is not exactly of the above sort insteadit is according to general theories for classical orthogonal polynomials [103] theequilibrium measure of the domain˝ (or its closure) In place of mother body termssuch as ldquomadonna bodyrdquo or electrostatic skeleton has been used for such cases see[21 50] for example Despite the different nature of the given body much of thetheory in Sect 62 below applies also to such kinds of skeletons

62 Some General Properties of Mother Bodies 81

62 Some General Properties of Mother Bodies

We collect here a few general facts which are known about mother bodies Someof these facts (more precisely Propositions 61 63 64) are taken from [35]where they are stated for D ˝ but in an arbitrary number of dimensions Forconvenience we repeat these results here including proofs

We start with a simple observation which will repeatedly be referred to

Lemma 61 If is a signed measure with compact support then U and C arelocally integrable functions Hence the distributional properties of these functionsdepend only on their values almost everywhere For example if U or C vanishesexcept on a null-set then necessarily D 0

The same applies to U13 if 13 is a compactly supported distribution of order atmost one

Proof Since the fundamental logarithmic kernel is locally integrable as well as itsfirst order derivatives the first statement in the lemma follows by a direct applicationof Fubinirsquos theorem So U and C are locally integrable functions which then fullyrepresent the corresponding distributions in the usual way

A distribution 13 of order at most one is locally a sum of distributional derivativesof a measures Assume for example considering only one term in such a sum that13 D

x where is a (signed) measure Then defining the potential of 13 as theconvolution with the fundamental solution ie U13 D U13ı D 13 Uı we see thatU13 D

x Uı D x Uı Here the last factor again has a locally integrable

singularity and the last statement of the lemma follows from Fubinirsquos theorem asbefore

Proposition 61 Let be a mother body for Then among (signed) measures 1313j satisfying just M1 and M3 we have that

(i) supp is minimal as a set if supp 13 supp then 13 D (ii) is maximal with respect to the partial ordering by potentials if U13 U

holds everywhere then 13 D (iii) is an extremal element in the convex set of measures satisfying M1 M3

if D 12131 C 132 then D 131 D 132

Proof In view of Lemma 61 and axiom M4 (for ) it is enough to prove that therelevant potentials agree in C n supp

So let D be a component of C n supp and set w D U13 U (cases i and ii)w D U13j U (case iii) We shall then show in all cases that w D 0 in D Weobserve that D ˝e curren by M5 (for ) and that w D 0 in D ˝e by M1 (for and 13 13j)

In case i w is harmonic in D and we then directly infer that w D 0 in all D byharmonic continuation

In case ii we have w 0 and w 0 in D Hence either w gt 0 in all D orw D 0 in all D But the first alternative has already been excluded so we again getw D 0 in D

82 6 Mother Bodies

In case iii we have supp 12131 C 132 D supp 131 [ supp 132 Hence supp 13j

supp and we are back to case iThe next proposition strengthens i in Proposition 61 by saying that supp is

minimal not only among supports of mother bodies but even among carriers of theanalytic functional associated to the body

Proposition 62 If is a mother body for then supp is a minimal carrier forthe analytic functional f 7 R

f dA f 2 O˝

Proof Assume that K supp is a carrier for f 7 Rf dA We shall then prove

that K D supp in other words that D 0 in C n KSince C D C outside˝ D supp C provides an analytic extension of Cj˝e

to C n supp By assumption there is also an analytic continuation ˚ of Cj˝e toC n K Clearly ˚ D C in C n supp (axiom M5 is crucial for this conclusion)But C ˚ both being locally integrable in C n K (C by Lemma 61 and ˚ becauseit is analytic) and supp being a null-set it follows that ˚ D C as distributions inC n K Since ˚ in analytic in C n K we conclude that D 0 there as desired

Mother bodies are not always unique for example non-convex polygons withuniform density always have more than one mother body as shown in [43] (whileconvex polygons have exactly one) It is easy to strengthen the axioms for a motherbody so that strict uniqueness is always achieved but then the price is even more rareexistence of mother bodies Let us give one theorem in this direction by introducingthe axiom

M6 supp does not disconnect any open set

which is much stronger than M5 The axiom says more explicitly that for anyconnected open set D the set Dnsupp is also connected Thus in a generic situationfor a mother body with supp consisting of isolated points and curve segmentsM6 rules out the curve segments The consequence for uniqueness is the following

Proposition 63 If and 13 are mother bodies for and satisfies M6 then13 D

Proof Keeping the notations from the proof of Proposition 61 we have that if 13satisfies M6 then supp 13 does not disconnect D Therefore w D 0 in D n supp 13 byharmonic continuation hence w D 0 ae in D which is enough for the conclusion

With the standard axioms M1ndashM5 one can at least argue that mother bodiesshould not occur in continuous families If t 7 t is a continuous family ofmother bodies for such that the derivative Pt exists in a strong enough sense thenone expects this derivative to be a distribution of order at most one since looselyspeaking it is the infinitesimal difference between two distributions of order zero(ie measures) Strictly speaking to be a distribution of order at most one meansto be something which can be written as a spatial derivative of a measure to be aderivative with the respect to an external parameter like t is not enough in generalBut under some additional assumptions one can connect the two kinds of derivatives

The following proposition is a rudimentary result on non-occurrence of continuousfamilies

Proposition 64 Assume that t 7 t is a family of mother bodies for some fixed which moves in the flow of a smooth vector field D z t in C Then the motherbodies are all the same t D 0 for all t

Proof That flows by can be taken to mean in differential geometric languagethat

tC L D 0

where L denotes the Lie derivative with respect to (see [24] for example) SinceL is a spatial derivative of first order L is a distribution of order at most onehence so is P by the equation

By Lemma 61 it now follows that U Pt 2 L1loc Clearly supp Pt supptSince Ut D U˝ in˝e by M1 for all t we conclude that U Pt D 0 in˝e Henceby harmonic continuation and M5 U Pt D 0 in all C n suppt ie almosteverywhere in C Thus Pt D 0 which is the desired conclusion

If is a mother body for then the quadrature formula

f dA DZ˝

f d f 2 O˝ (64)

holds In fact this is just a weaker version of (63) In particular and representthe same element in H ˝ In the definition of a mother body there is norequirement that shall have compact support in ˝ The reason is that one wantspolygons and other domains with corners to admit mother bodies

In the other direction even finite quadrature domains in the sense (341) do notalways admit mother bodies in the strict sense of Definition 61 and with D ˝ Indeed if

f dA DmX

nk1XjD0

ckjf jak (65)

then a necessary condition that D ˝ shall have a mother body is that nk D 1 andck0 gt 0 for all k But even that is not sufficient it may still happen that axiom M2fails for the only available candidate of mother body namely

ck0ıak

Example 61 Let f W D ˝ be a conformal map which is an entire function withf 0 D 0 Consider ˝ as a body with density one ie take D ˝ If f D a

84 6 Mother Bodies

then ˝ is a disk which is a quadrature domain and for which D a2ı0 is theunique mother body If f is a polynomial of degree 2 then ˝ is still a quadraturedomain with a multiple quadrature node at the origin but it has no mother bodysatisfying all of M1ndashM5 Indeed M2 and M3 fail

If f is not a polynomial (but still entire) then ˝ is not a quadrature domainand there is no mother body But the analytic functional in (347) still has thesingleton f0g as a carrier However there will be no distribution sitting exactly onf0g representing M only a hyperfunction (a Dirac multipole of infinite order)

Example 62 If a gt 0 is sufficiently large then there exists a quadrature domain ˝admitting the quadrature identity

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

One may view the right member as something of the formR

f d but then will bea complex measure which mixes the real and imaginary parts of f As an analyticfunctional f 7 R

˝f dA has f1 0C1g as a minimal carrier but D ˝ does not

have any mother body (eg M2 M3 will be violated for )One can still find real-valued representing distributions (with small support) but

then one need to connect the points ˙1 For example writing f D uC iv and setting D Œ1C1 we have

i f 1C f C1 D iZ

xdx D i

Taking real parts of (66) therefore givesZ˝

u dA D au0CZ

Here u runs over real-valued harmonic functions but of course complex-valuedfunctions u are allowed as well and then nothing prevents u from being analyticitself Hence we have a real-valued representation of the analytic functional f 7R˝

f dA by a distribution of order one with support on the line segment DŒ1C1

63 Reduction of Inner Product to Mother Body

What is special with the inner product in Ha˝ is that it in contrast to the L2-innerproduct in the Bergman space L2a˝ immediately collapses to any mother body for D ˝ Indeed using (64) gives

h f gi D 1

Hzwf zgw dzdw (67)

63 Reduction of Inner Product to Mother Body 85

Choosing here f D kz g D kw for zw 2 ˝e and using (338) gives

1 Fzw D hkz kwi D 1

Hu vdu

Recall that kz D 1= z and that Fzw denotes the restriction of Ezw to˝e ˝e including analytic continuations of this Thus we see clearly that Fzwhas an analytic continuation to C n supp C n supp if is a mother body

Similarly Fzw extends to C n K C n K whenever K ˝ is a carrier forf 7 R

˝f dA This statement can be sharpened to become a full-fledged assertion on

regularity of free boundaries applying for example to the classical obstacle problem[75] We give some details on this in Sect 64 below

The above can be equivalently expressed with the inner product written on theform (325)

h f gi D 1

1 Fzwf zgw dzd Nw (68)

Here we have replaced the original Ezw in (325) equivalently (328) by itsanalytic continuation Fzw and then the path of integration can be shrunk tosurround just the support of any mother body with the mother body itself then beingrepresented by the jump of Fzw across the arising slits or by residues of polesif Fzw has point singularities In the integrals above f g 2 Ha˝ but forsimplicity we may think of f and g as being good analytic functions say in O˝

In terms of the Schwarz function see (220) the analytic continuation of Fzwis connected to Hzw by (224) ie by

Fzw D z SwSz NwHzw zw 2 ˝ n supp

Similarly is related to the singularities of Sz in a direct way eg simple polescorrespond to point masses and jumps between different branches give line sourcesexplicitly by formulas like (621) below Using these relationships the two forms ofthe inner product (67) and (68) can be identified

We remark that the information in the right member of (67) having both Hzwand present is actually redundant The function Hzw contains all informationof ˝ and hence of all mother bodies and in the other direction if is known andsatisfies at least axioms M1 M2 in the definition of a mother body then ˝ canbe reconstructed by a partial balayage as mentioned in Sect 61

So Hzw and are coupled to each other even though in a rather implicitway The redundancy may be explained in terms of moments (see Sect 22) asfollows complete knowledge of Hzw (or Ezw) is equivalent to knowledgeof all complex moments Mkj whereas knowledge of just means knowledgeof the harmonic moments Mk D Mk0 Indeed the complex moments are thecoefficients of the expansion of Ezw at infinity while the harmonic momentsare the coefficients of C˝z D Cz at infinity More specifically contains

86 6 Mother Bodies

information of singularities in a fairly explicit way while the remaining informationin Hzw can be considered to be of a more lsquosoftrsquo character

Part of the above discussion is also relevant the Bergman inner product whenthis is written on the form (327) Indeed in terms of Fzw this becomes

f gL2˝ D 1

log Fzwf zgw dzd Nw (69)

If F has an analytic continuation then also log F has it to a certain extent Howeverthis may become obstructed by branch-cuts because the logarithm is multivalued(and note that Fz z D 0 on ˝) Such branch cuts at least prevent the innerproduct from collapsing when˝ is a quadrature domain and in general they seem tohave the effect that zeros of Bergman orthogonal polynomials are less willing to godeeply into the domain compared to what is the case for exponential polynomials

64 Regularity of Some Free Boundaries

This section contains an outline of ideas behind the main result in [36] which isa theorem on analyticity of some free boundaries namely those boundaries whichadmit analytic continuation of the Cauchy transform The important point is that itassumes no a priori regularity of ˝ ˝ is from outset just a bounded open setHowever we shall here (for briefness) simplify matters a little following partly[38] but full details are given in [36] The statement when formulated in terms ofthe Cauchy kernel kz D 1= z is as follows

Theorem 61 Let ˝ C be a bounded open set and let K ˝ be compact Thenthe following assertions are equivalent

(i) The map C n˝ C given by

z 7 hkz 1i

extends analytically to C n K C(ii) The map C n˝2 C given by

zw 7 hkz kwi

extends analytically to C n K2 C(iii) The Hilbert space-valued map C n˝ H ˝ given by

z 7 kz

extends analytically to C n K H ˝

64 Regularity of Some Free Boundaries 87

Recall that in terms of Cauchy and exponential transforms hkz 1i D C˝zhkz kwi D 1 Ezw As for the Hilbert space-valued map we say that a map˚ W C n K H ˝ is analytic if and only if the function z 7 h˚z hi is analyticfor each h 2 H ˝ The three assertions say more precisely that there exist analyticfunctions f W C n K C F W C n K2 C ˚ W C n K H ˝ such thatin the respective cases hkz 1i D f z for z 2 C n ˝ hkz kwi D 1 Fzw forzw 2 C n˝ kz D ˚z for z 2 C n˝

Proof It is easy to see that iii ) ii ) i so we just prove that i ) iiiWe make the simplifying assumptions that (31) and (61) hold but as shown in [36]these assumptions can eventually be removed

Assuming then i let f z denote the analytic continuation of hkz 1i D C˝zto C n K C and we shall find an analytic continuation ˚z of kz itself It isconvenient to continue f further in an arbitrary fashion over K and to all C andthen think of f as the Cauchy transform of something living on K This somethingcan be taken to be a signed measure or even a smooth function if we just enlargeK a little This does not change anything in principle since this enlargement can bemade arbitrarily small and we may keep the notation K

Thus we assume that after the extension D fNz is a smooth function in C with

supp K (610)

This means that the assumption i takes the form

C˝ D C on C n˝ (611)

equivalently

hkz 1i D hkz 1i for z 2 C n˝

and we claim then that the analytic extension of kz itself is given by

˚z D kz (612)

Similarly the continuation of hkz kwi in ii of the theorem will be given by

1 Fzw D hkz kwi D h˚z ˚wi

That ˚ defined by (612) is in fact analytic as a map C n K H ˝ isimmediate from the definition of analyticity and from (610) So it just remainsto verify that ˚z D kz as elements in H ˝ for z 2 C n˝ There is an beautifulapproximation theorem [8] saying that for a completely arbitrary bounded open set˝ the finite linear combinations of the Cauchy kernels with poles outside ˝ aredense in the space of integrable analytic functions in ˝ The theorem is based onan ingenious construction due L Bers and L Ahlfors of mollifiers which exactly

88 6 Mother Bodies

fit the modulus of continuity for Cauchy transforms The theorem can be adapted toharmonic and subharmonic functions as well (cf [56]) and has in such forms beenof crucial importance in the theory of quadrature domains from the work of Sakai[89] and onwards A particular consequence is that the statement (611) is equivalentto the quadrature identity

h dA DZ

Kh dA (613)

holding for all integrable analytic functions h in ˝ Choosing in (613) the integrable analytic function

h DZ˝

Hzwkzw dAw

where z 2 ˝e and 2 L1˝ gives

hkz i D hkz i

Thus since L1˝ is dense in H ˝ kz D kz as elements in H ˝ for z hellip ˝ and by continuity also for z hellip ˝ This proves the theorem

It follows from the definition (21) of the exponential transform that Ez z D 0

for z 2 ˝ unless this is a point of zero density for˝ When the Cauchy transformC˝z and hence by Theorem 61 E˝zw has an analytic continuation across˝ then ˝ has positive density at all points of ˝ see [36] It therefore follows thatthe analytic continuation Fzw of Ezw in this case becomes a good definingfunction for ˝

Corollary 61 If the Cauchy transform of a bounded open set ˝ has an analyticcontinuation from the exterior of ˝ across ˝ to a compact set K ˝ then ˝ isnecessarily contained in a real analytic variety

˝ fz 2 C n K W Fz z D 0g

The first complete proofs of analyticity of boundaries admitting analytic con-tinuation of the Cauchy transform or boundaries having a one-sided Schwarzfunction were given by Sakai [92ndash94] For overviews of the regularity theory offree boundaries in general see eg [25 75] In a different context the exponentialtransform has much earlier been used implicitly in characterizations of boundariesof analytic varieties in C2 See [4 54]

65 Procedures for Finding Mother Bodies 89

65 Procedures for Finding Mother Bodies

Let be the density of a mass distribution positive and bounded in a domain ˝ zero outside ˝ and we shall describe some procedures for finding candidates ofmother bodies for For these to work one has to assume that the density is realanalytic in ˝ and that the boundary ˝ is at least piecewise analytic Some of themethods are implicit in the general theory of quadrature domains as presented forexample in [89 101] while other (those based on jumps of the Schwarz function)have been elaborated in [95] and similar papers

Reasoning backwards assume first that we already have a mother body whichreaches ˝ only at the non-smooth points (the ldquocornersrdquo) Then the function

u D U U (614)

is non-negative and satisfies

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

In particular away from supp in ˝ we have

u D (617)

Going now in the other direction if is not known then we may start by tryingto solve the Cauchy problem (617) (616) in a neighborhood (in ˝) of the analyticparts of ˝ Unique local solutions do exist in view of the Cauchy-Kovalevskayatheorem Assuming that the domain of definitions of these local solutions cover allof ˝ we may then try to glue these pieces to a continuous function u in ˝ This uwill satisfy (616) and then we may simply define by (615) In lucky cases this will then be a mother body

An alternative but related procedure uses the Schwarz function Sz If D 1

in ˝ then the relationship between u and Sz is in one direction

Sz D Nz 4u

z (618)

and in the other direction

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

Here z0 2 ˝ and the relation holds in a neighborhood of that analytic piece of˝ to which z0 belongs The function u in (618) is sometimes called the modifiedSchwarz potential see [101]

90 6 Mother Bodies

In the general case one may first choose a fixed function ˚ satisfying

ie ˚ satisfies (617) alone with no concern of boundary data Since all data arereal analytic we shall write and ˚ as D z Nz and ˚ D ˚z Nz with analyticdependence in each argument When D 1 the natural choice of˚ is˚z Nz D 1

As one easily checks the relationship between u and Sz in general is

z˚z Sz D z˚z Nz u

which replaces (618) but only gives Sz implicitly from u and in the other direction

uz D ˚z Nz ˚z0 Nz0 2ReZ z

z˚ Sd (620)

In these equations z˚ denotes the derivative of ˚ with respect to the first variableHere z0 2 ˝ and (620) holds in a neighborhood of z0 Compare [95] and also[94]

To find candidates for mother bodies one continues the branches of the Schwarzfunction as far as possible and then try to glue them However since Sz isanalytic one can never glue it to become continuous (because then it would beanalytic throughout ˝ which is impossible) In the generic case there will bejump discontinuities along line segments like branch cuts in the algebraic casePole singularities can also occur The mother body then comes from the jump ofSz across these line segments (plus residues for poles) the contribution from anoriented arc being

d D 2iŒz˚z Szjump dz along (621)

If s is an arc length parameter then replacing dz by dz=ds above gives the density of with respect to s along the arc The jump is defined as the value on the right-handside minus the value on the left-hand side The measure should be positive or atleast real and that requirement singles out at each point z 2 ˝ only finitely manypossible directions for a branch cut passing through the point the requirement forsuch a direction being

Re Œz˚z Szjump dz D 0 along

See further discussions in [95] On squaring (621) the requirement that shall bereal translates into a certain quadratic differential being asked to be positive Thusthe theory connects to the theory of trajectories of quadratic differentials (see [105])a direction which has been pursued in a random matrix context by Bleher and Silva[9]

We next give an example showing that the good property of carriers that theintersection of two convex carriers is again a (convex) carrier is not shared bysupports of mother bodies The example is inspired by the construction by Sakai[87] of two simply connected Jordan domains which have the same set of harmonicmoments

Example 63 Choose numbers 1 lt r lt R lt 1 and consider the two annuli

(AC D fr lt jz C 1j lt RgA D fr lt jz 1j lt Rg

Considered as bodies of density one they have the same potentials in the exteriorregions as the point masses R2r2ı1 and R2r2ıC1 respectively It followsthat the two mass distributions AC

CR2 r2ıC1 and ACR2 r2ı1 have

the same potentials far away say for jzj gt R C 1Now we disturb the above by replacing the first terms by ACnA

and AnAC

respectively This means that we remove some small pieces from the annuli butwhat is important is that we remove the same piece (namely AC A) from bothTherefore the two new mass distributions

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

still have the same potentials for jzj gt R C 1The next step is that we construct mother bodies C and of ACnA

andAnAC

This is easily done using the procedures shown in Sect 65 For examplethe potential u in (614) is given as the minimum of the solutions of (615) (616)connected to the different boundary segments Specifically setting

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

for the solution of u D 1 in C u D jruj D 0 on jzj D R (and similarly for littler) we define u˙ by

(uCz D minACnA

fuRz 1 urz 1 uRzC1 urzC1g for z 2 AC n Auz D minAnAC

fuRz 1 urz 1 uRzC1 urzC1g for z 2 A n AC

Thenu˙ D 1 except on some sets where two of the potentials minimized over areequal where we get distributional contributions (negative measures) Defining then

(C D 1 uC D 1 u

we have Bal C 1 D ACnA Bal 1 D AnAC

92 6 Mother Bodies

Now C C R2 r2ıC1 and C R2 r2ı1 have the same exteriorpotentials as C and respectively Finally we sweep these mass distributions tobodies of constant density by partial balayage To fit our general setting we shouldhave density one but then we may need to first blow up the mother bodies by afactor t gt 1 The result can be written as

(Bal tC C R2 r2ıC1 1 D Bal tC 1 D ˝Ct

Bal t C R2 r2ı1 1 D Bal t 1 D ˝t

If here t gt 1 is sufficiently large then ˝Ct D ˝t This follows fromgeneral theory of partial balayage or quadrature domains Indeed when t is large˝˙ will necessarily be simply connected in fact essentially circular as one realizesby various comparison arguments or by the sharp ldquoinner normal theoremrdquo [44 45]And then the equality˝Ct D ˝t follows from uniqueness theorems originallydue to Sakai [89] (for example Corollary 92 in that text)

The conclusion of all this is that the domain ˝ D ˝Ct D ˝t has twodifferent mother bodies tC C R2 r2ıC1 and t C R2 r2ı1 and itis easy to see that the are not only different even the convex hulls of their supportsare different

Remark 61 Some previously known examples of domain having different motherbodies like Zidarovrsquos example [113] have the drawback in this context that theconvex hulls of their supports coincide

Chapter 7Examples

Abstract In this chapter we present the results of our numerical experimentsconcerning zeros of exponential polynomials with pictures indicating asymptoticdistributions (or at least the distribution of the first 50ndash100 zeros) We also makethe corresponding theoretical construction of mother bodies and compare the twoIn many cases there are reasonable but far from perfect agreements between thesets where zeros seem to accumulate and the supports of the corresponding motherbodies And here we reach terra incognito as there is no agreement between thedensities they seem rather to be complementary to each other In one case we havecomplete results with theoretical proofs namely for the ellipse Taking a standardellipse with foci ˙1 the zeros go to the focal segment with density proportional to1=

p1 x2 1 x 1 while the mother body for the ellipse with uniform mass

distribution has densityp1 x2 on the same segment

71 The Unit Disk

For the unit disk Hzw D 11z Nw and HaD degenerates to the one dimensional

space with inner product

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

henk ersi D(1 if n k D r s 0

0 otherwise

It follows that H D is generated by the anti-analytic functions fe00 e01 e02g Df Nzk

kC1 W k D 0 1 2 g and that this is an orthonormal basis It also follows thate00 D e11 D e22 D e01 D e12 D e23 D etc as elements in H D

94 7 Examples

Note that on the other hand Hzw coincides in this case with Szegoumlrsquos kernelthat is the reproducing kernel of the Hardy space associated to the unit disk

72 The Annulus

For the annulus

˝ D fz 2 C W r lt jzj lt Rg

we have E˝zw D ED0R=ED0r which by (25) gives

H˝zw D z Nwz Nw r2R2 z Nw zw 2 ˝

Gzw D(

zr2z Nw r lt jzj lt R jwj gt R

zR2z Nw r lt jzj lt R jwj lt r

The analytic space Ha˝ is generated by the powers zn z 2 Z and taking thenorm into account gives the orthonormal basis fen W n 2 Zg defined by

enz D8lt

R2r2 n lt 0

R2r2 n 0

We can truncate the multiplier Z by the coordinate on the finite dimensionalsubspaces generated by zk n k n The result will be a weighted shift Zn withzero on the main diagonal and only the lower sub-diagonal non-null In particularZn is nilpotent and hence the counting measure n of its spectrum is simply ı0In conclusion all exponential orthogonal polynomials are monomials and the onlylimit measure of their zeros is ı0

Since ˝ is not a quadrature domain feng is actually an orthonormal basis in allH ˝ (see Theorem 31) Given an arbitrary f 2 H ˝ we can therefore expand

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

In the hope of being able to consider f as an analytic function we insert the variablez and convert the expansion into a power series

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

73 Complements of Unbounded Quadrature Domains 95

Here the first term converges for jzj gt R lim supn1 jnj

pjcnj and the second termfor jzj lt r= lim supn1 jnj

pjcnj The lim sups here are 1 but nothing moreis guaranteed Taking for example cn D 1=jnj the first term in the expansion off z converges for jzj gt R and the second term for jzj lt r hence the whole seriesconverges nowhere

The orthonormal expansion (71) gives a one-to-one identification betweenelements f in H ˝ and `2Z-sequences of coefficients cn Therefore the corre-sponding power series expansion (72) gives a good picture of the nature of elementsin H ˝ from an analytic point of view they are simply formal power serieswith coefficients in an `2-space but with possibly empty convergence region So ingeneral there is no way to associate a value at any point for a ldquofunctionrdquo f 2 H ˝In particular there is no reproducing kernel for H ˝ If there had been this wouldhave been given by

enzenw zw 2 ˝

(cf (340)) but this series converges nowhereThe annulus ˝ has a unique mother body sitting with uniform density on a

circle jzj D for certain in the interval r lt lt R namely that value for whichur D uR where ur uR are defined by (622) But as is clear from the abovenot even on this circle there are any continuous point evaluations for elements inH ˝

73 Complements of Unbounded Quadrature Domains

Let D be a quadrature domain with 0 2 D The inversion z 7 1=z then takes thecomplement of D to a bounded domain

˝ D inv De D fz 2 P W 1z

2 P n Dg

Since D is bounded we have 0 2 ˝ The exterior ˝e D inv D is what is calledan unbounded quadrature domain see in general [63 70 88 93 99 100] for thisnotion We shall not need much of the theory of unbounded quadrature domains wejust remark that the class of bounded and unbounded quadrature domains as a wholeis invariant under Moumlbius transformations (see in particular [100] and [93])

Let d be the order of the quadrature domain D Then by (344) the exteriorexponential transform of D is of the form

EDzw D FDzw D Qz NwPzPw

jzj jwj gtgt 1

96 7 Examples

where Pz a monic polynomial of degree d and Qz Nw is a polynomial of degreed in each of the variables Some more details on the structure of Qz Nw can beread off from (48) By using Lemma 21 together with (224) or more directly byformula (314) in [36] one obtains from FDzw the interior exponential transformof ˝

H˝zw D CQ 1z 0Q01Nw

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

Here Sz D SDz is the Schwarz function for D that branch which is single-valued meromorphic in D and equals Nz on D In addition C D H˝0 0 and p qare polynomials satisfying p0 D q0 0 D 1 Specifically

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

Note that 1z S0 is a factor in Q 1z 0 because QS0 0 D 0 It follows that pzis a polynomial of degree at most d 1

Since Hzw is positive definite the zeros of pz must be outside ˝ This factcan also be proved directly by using arguments with the Schwarz function includinga valency statement in [101] It also follows that 1=qzw is positive definite in ˝

If D is connected and has m boundary components (m 1) the algebraic curveloc Q W Qzw D 0 has genus m 1 In fact this curve is (after completionin projective space P2 and resolution of singularities) canonically isomorphic tothe compact Riemann surface (Schottky double of D) obtained by completing Dwith a backside (a copy of D provided with the opposite structure) the two piecesglued together along D On the frontside the isomorphism is given by D 3 z 7z Sz 2 loc Q Recall that QDz Sz D 0 identically From this we obtain thetotal number of branch points of Sz considered as an algebraic function IndeedSz has d branches over the Riemann sphere and the Riemann-Hurwitz formula[22] shows that the number of branch points is

b D 2m C d 2 (75)

Turning to ˝ D inv De the Schwarz function for ˝ is related to that for Dby

S˝z D 1

and it has a single-valued branch in ˝e namely that branch which equals Nz on ˝ However continuing that branch into ˝ it will not be single-valued (unless d D 1)

Instead its structure of branch points will determine the mother body structure anymother body will consist of arcs which end up in these branch points

Notice that in the present situation all poles of S˝z are located outside ˝ soany mother body for˝ will consist solely of branch cuts and from (75) we see that2m C d 2 is an upper bound for the number of end points of these cuts

731 The Ellipse

The ellipse can be obtained as the complement of the inversion of a certain two pointquadrature domain known as the Neumann oval see [47 69 93 101] It is also aspecial case of the hypocycloid to be treated in the next subsection The values ofthe parameters in (75) are then d D 2 m D 1 b D 2

The standard ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

with half axes a gt b gt 0 has a unique mother body [102] it is the measuresupported by the focal segment Œc c (c D p

a2 b2 gt 0) given by

dx D 2ab

c2 x2 dx c lt x lt c

(For some higher dimensional aspects see [64]) The madonna body (skeleton forthe equilibrium measure) 13 for the ellipse has the same support but a differentdensity

d13 D dx

c2 x2 c lt x lt c (76)

The Schwarz function for the ellipse is

Sz D a2 C b2

c2z 2ab

pz2 c2

that branch of the square root chosen which is positive for large positive values ofz The interior exponential transform Hzw is obtained from (73) with qz Nz D1 x2

b2(z D x C iy) and where pz turns out to be constant see also Sect 732

in this respect Specifically this gives

Hzw D C

4a2b2 C a2 b2z2 C Nw2 2a2 C b2z Nw

where C D 4a2b2H0 0 gt 0

98 7 Examples

It is known that the Bergman orthogonal polynomials have 13 as the unique weaklimit of their counting measures The same turns out to be true for the exponentialpolynomials

Proposition 71 Let 13n be the counting measures for the zeros of the exponentialpolynomials for the ellipse Then in the above notations

13n 13

as n 1

Proof The proof is based on hyponormal operator theory and goes as follows LetS 2 L `2N denote the unilateral shift operator

Sek D ekC1 k 0

where ek stands for the orthonormal basis of `2N Let 0 lt r lt 1 be a parameterand consider the operator T D rS C S Since ŒSS D e0 ˝ e0 we infer

ŒTT D ŒrS C S rS C S D 1 r2ŒSS D 1 r2e0 ˝ e0

Since the operators S and S commute modulo compact ones the essentialspectrum of T is equal to the set

essT D fr C 1

jj D 1g

that is it coincides with the ellipse of semiaxes 1˙ r Thus the spectrum of T is thesolid ellipse The focal segment for this ellipse is Œ2pr 2

pr (Fig 71)

The finite dimensional subspaces to compress T on are

HnT e0 D spanfe0 e1 en1g

and the associated truncated operators are

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

This is a Jacobi matrix and it is well known that its characteristic polynomial isup to a normalizing factor equal to the Chebyshev polynomial of the second type

Fig 71 Zeros of the orthogonal polynomials Pn n D 10 20 30 for an ellipse

r Recall that

Uncos D sinn C 1

so that indeed the zeros of Unz

r asymptotically distribute as in (76) ie

according to the probability distribution

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

A hypocycloid is a curve traced out from a point in a disk when this disk rolls insidea larger circle In case the point lies on the boundary of the disk the hypocycloidhas outward pointing cusps otherwise it is a smooth curve If the ratio between theradii of the two circles is an integer then the curve obtained is a closed curve andbounds a domain We shall consider such a hypocycloid domain ˝ centered at theorigin

Elementary considerations show that if the distance from the origin to the centerof the smaller disk is a gt 0 and the moving point zt sits on distance b gt 0 fromthe center of this smaller disk then the motion is given by

zt D aeit C beid1t

100 7 Examples

Here d gt 1 is the ratio between radius of the larger circle to that of the smaller oneWe assume that d is an integer so d 2 The geometry of the above configurationworks out only if

a d 1b (77)

The equality case here corresponds to the hypocycloid being singular with cuspsSetting D eit above we see that the hypocycloid simply is the image of the unit

circle under the rational function

D a C b1d

In addition (77) is exactly the condition for to be univalent in De Thus is

then a conformal map De ˝e subject to standard normalization at infinity (in

particular 1 D 1)After inversion z 7 1=z the exterior domain˝e turns into a bounded domain D

with conformal map W D D given by

a C bd

Since this is a rational function of order d D is a quadrature domain of order d Thed poles are located at the dth roots of a=b In particular we see that the multiplepole of order d 1 of at the origin has split into d simple poles for Since Dis a (bounded) quadrature domain ˝e is an unbounded quadrature domain with amultiple quadrature node at infinity

The number d above has the same meaning as in the beginning of this Sect 73and the number of boundary components of D or ˝ is m D 1 The boundary of Dis given by an equation

D W Qz Nz D 0

where the polynomial Qzw has degree d in each of the variables hence theboundary of ˝ is given by qz Nz D 0 with qzw as in (74) The total numberof branch points of each of SDz and S˝z is given by the Riemann-Hurwitzformula (75) ie equals b D 2m C d 2 D 2d 1

In addition to these branch points the curve loc Q has singular points Takingmultiplicities into account the total number of singular points is d 12d 1according to the genus formula [29] The quadrature nodes account for dd 1

singular points located at points of infinity of the two dimensional projective spaceP2 Further on there are d C 1 singular points visible in the real as isolated rootsz 2 C of Qz Nz D 0 or cusps on the boundary in the cusp case One of these pointsis the origin (see argument below) the point of inversion the remaining d pointsare located in D (or on D in the cusp case) on those radii from the origin whichare intermediate between the radii leading to the quadrature nodes The remaining

d12d1dC1D dd3 are not visible in the real The above statementsas well as some assertions below are elaborated in detail and proved [33]

Using that Qz Nz must be invariant under rotations generated by z 7 e2 i=d onefinds that Qzw more exactly is of the form

Qzw D ˛0 C ˛1zw C ˛2z2w2 C C ˛d1zd1wd1 C ˛dzdwd C ˇzd C Nwd

Because we have assumed that D is simply connected it also follows from potentialtheoretic consideration together with symmetries that ˛0 D 0 Indeed the functionu in (618) must have a stationary point at the origin hence SD0 D 0 and soQ0 0 D 0 And since Qz Nz 0 in D it follows that the origin is a singular pointon loc Q By normalization we also have ˛d D 1

Turning to qzw and pz see (74) it follows that

qzw D 1C ˛d1zw C ˛d2z2w2 C C ˛1zd1wd1 C ˇzd C ˇwd

where we have used that ˇ is real and

pz D zdQ1

z 0 D ˇ

In general pz has degree d 1 as mentioned after (74) and when the origin isa singular point it has always degree d 2 as a consequence of Q0 0 D 0 Butin the present case it is even better than so pz is simply a constant This makes theinterior exponential transform (73) to appear on the very simple form

H˝zw D C

1C ˛d1z Nw C ˛d2z2 Nw2 C C ˛1zd1 Nwd1 C ˇzd C ˇ Nw d

Since we started out from having uniformizations of the curves Qzw D 0 andqzw D 0 for example

q 1= N D 0 2 P (79)

we can easily determine the coefficients in (78) in terms of the coefficients a and bof The result for d D 3 is

H˝zw D C

a2bz3 C Nw3 b2z2 Nw2 a2b2 C a4 2b4z Nw C a2 b23

where we have changed the normalization in the denominator hence changed thevalue of C For any d 2 the values of the coefficients ˛j ˇ are obtained by solvinga linear system of equations obtained from the substitution (79) So this is quitestraight-forward Of course one can also find the polynomial qzw by classical

102 7 Examples

elimination theory without knowing anything a priori about qzw but that leadsto more difficult equations to solve

For d D 2 we are in the ellipse case discussed in terms of different methods andnotations in Sect 731 and in that case we could find the Schwarz function and themother body explicitly But already in the case d D 3 it is not so easy anymoreIt is difficult to write down the Schwarz functions SDz and S˝z explicitlysince this would involve solving a third degree algebraic equation However thelocations of the b D 2d 1 D 4 branch points can be estimated For examplefor S˝z the branch points are inside ˝ one at the origin and the other threeon the radii corresponding (under the inversion) to the radii just mentioned Thusany mother body will consist of arcs connecting these points and analysis of therequirements M1 M5 in Chap 6 leaves only one possibility open namelythat its support consists of the straight lines (radii) from the origin to the threebranch points outside the origin Figure 72 indicates that zeros of the exponentialpolynomials do converge to this configuration

Similar results have been obtained for other sequences of polynomials see forexample [20 55] and for a warning [85]

Fig 72 Zeros of the orthogonal polynomials Pn n D 60 70 80 for the hypocycloid defined by D 2 C 1=82

74 Lemniscates 103

74 Lemniscates

For R gt 0 we consider the lemniscate

˝ D fz 2 C W jzm 1j lt Rmg

Thus the boundary is given by

zm 1Nzm 1 D R2m

which on solving for Nz gives the Schwarz function

Sz D m

szm 1C R2m

zm 1 (710)

the appropriate branch chosen Obviously˝ is symmetric under rotations by angle2=m

We start by computing the mother body There are three cases to consider

1 The disconnected case 0 lt R lt 1 ˝ then consists of m ldquoislandsrdquoBranch points for Sz z D 1 and 1 R2m1=m (the positive root) plus rotationsby 2=m of this The branch cuts giving rise to a real actually positive motherbody are the interval Œ1R2m1=m 1 ˝ plus the rotations of this The motherbody itself is

dx D sin m=

xm 1C R2m

xm 11=m dx 1 R2m1=m lt x lt 1

plus rotations by 2=m2 The critical case R D 1 ˝ still consists of m islands meeting at the origin

which now is a higher order branch point The branch cuts are 0 1 ˝ plusrotations and the formula for d is the same as above

3 The connected case 1 lt R lt 1In this case the only branch points contained in˝ are the mth roots of unity andthe branch cuts giving a real mother body are the radii from the origin to theseroots of unity The mother body itself is

dx D sin=m

xm 1C R2m

1 xm1=m dx 0 lt x lt 1

plus rotations

104 7 Examples

Fig 73 Zeros of the orthogonal polynomials for lemniscate with m D 3 and R D 09

These mother bodies seem to agree as far as the supports are concerned withlimits of counting measures found in numerical experiments (Fig 73)

One may compare with what is known for the Bergman polynomials In that case should be replaced by the equilibrium measure which is related to the exteriorGreenrsquos function and is easy to compute Indeed the Greenrsquos function is

g˝ez1 D 1

mlog jzm 1j log R

the equilibrium measure sits on the boundary and its density with respect to arclength is proportional to the normal derivative of the Greenrsquos function Precisely dA is to be replaced by the measure

Rmjdzj on ˝

75 Polygons 105

The unique mother body (ldquomadonna bodyrdquo) is in terms of the harmonic continua-tion of the above Greenrsquos function to all C

2g˝e1 D 1

where D e2 i=mThe comparison we wish to make here is first of all that the madonna body

is completely different from the mother body for D ˝ and second thatfor the counting measures for the Bergman polynomials there is rather completeknowledge saying that certain subsequences converge to the madonna body butother subsequences (actually most of them) converge to an equilibrium measurefor a sublemniscate namely jzm 1j lt R2m (at least when 0 lt R lt 1) Forour exponential polynomials the entire sequence of counting measures seems toconverge to a measure having the same support as the mother body

75 Polygons

751 Computation of Mother Body

For convex polygons with D 1 in ˝ it is known [34] that

uz D 1

2dist z˝e2

for the unique mother body which sits on the ldquoridgerdquo ie the set of those points in˝ which have at least two closest neighbors on the boundary On each line segmentin the ridge the density of with respect to arc length is linear and it goes down tozero where the ridge reaches ˝ namely at the corner points

As for the supports of these mother bodies they are in reasonable but not perfectagreement with the numerical findings for the limits of the counting measures Thedensities do not agree however

752 Numerical Experiments

Asymptotic zeros for a square rectangle pentagon and hexagon are depicted inFigs 74 75 76 77)

106 7 Examples

Fig 74 Zeros of the orthogonal polynomials Pn n D 40 50 75 for the square with vertices atthe forth roots of unity

Fig 75 Zeros of the orthogonal polynomials Pn n D 60 70 80 for the 2 1 rectangle withvertices at 2C i 2C i 2 i and 2 i

75 Polygons 107

Fig 76 Zeros of the orthogonal polynomials Pn n D 100 110 120 for the canonical pentagonwith vertices at the fifth roots of unity

Fig 77 Zeros of the orthogonal polynomials Pn n D 80 90 100 for the canonical hexagon withvertices at the sixth roots of unity

108 7 Examples

76 The Half-Disk and Disk with a Sector Removed

Let ˝ be the half-disk

The modified Schwarz potential is

u D minu1 u2

where uj solves the Cauchy problem (616) (617) with D ˝ for respectivelythe straight ( j D 1) and circular ( j D 2) part of the boundary

(u1z D 1

2Re z2

u2z D 14jzj2 log jzj2 1

It follows that the equation for the support of the mother body is

x2 y2 C logx2 C y2 D 1

This agrees almost exactly with where the zeros of the orthogonal polynomials arefound in numerical experiments See Fig 78

Considering a more general convex circular sector say

˝ D fz 2 C W jzj lt 1 j arg zj lt ˛g

where 0 lt ˛ lt 2

there will be three branches of u corresponding to the differentpart of ˝ to match the straight line segment in the first quadrant the circular arcand the line segment in the fourth quadrant contribute respectively with

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

76 The Half-Disk and Disk with a Sector Removed 109

Fig 78 Zeros of theorthogonal polynomials Pnn D 100 110 120 for thehalf-disk

The particular choice ˛ D 4

results in the explicit expressions

u1z D 14x2 C y2 2xy

u2z D 14x2 C y2 logx2 C y2 1

u3z D 14x2 C y2 C 2xy

The modified Schwarz potential will in the above cases (any convex sector) be u Dminu1 u2 u3 and the presumably unique mother body D 1 u

Finally we may consider a non-convex sector say

˝ D fz 2 C W jzj lt 1 j arg zj lt 3

The system (711) is then modified to

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

Fig 79 Zeros of the orthogonal polynomials Pn n D 60 80 100 for the 3=4-disk

It is still possible to glue u1 u2 u3 to one continuous function u in ˝ but thecorresponding measure D 1 u will no longer be positive it will be negativeon a line segment of the positive real axis starting at the origin Thus it is out ofquestion that can be the weak limit of some counting measures

There are however two (positive) measures which do satisfy all requirementsof a mother body These are obtained by decomposing the ˝ into a half disk plusa quarter disk and adding the mother bodies for these This can be done in twodifferent ways but none of the mother bodies obtained come close to agreeing withthe results of numerical experiments These are shown in Fig 79

762 Numerical Experiment

See Figs 78 and 79

77 Domain Bounded by Two Circular Arcs 111

77 Domain Bounded by Two Circular Arcs

Consider a domain˝ bounded by two circular arcs with centers on the real axis andmeeting at the points ˙i Denoting by a b 2 R the centers assumed distinct theequations of the circles are

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

The corresponding solutions ua ub of the local Cauchy problem (616) (617) with D 1 ie the functions given by (619) are

uaz D 1

4jz aj2 1C a2log jz aj2 C 1 log1C a2

similarly for ub Note that ua satisfiesua D 11Ca2ıa in all C ua D rua D 0

on Ca (similarly for ub)Since a curren b we may assume that a lt b and also that a lt 0 It is easiest to

think of the symmetric case that b D a but the analysis below does not reallyrequire that It is easy to see that given a and b there are four different domains ˝such that ˝ Ca [ Cb With Da Db denoting the open disks bounded by Ca Cb

respectively these domains are Da Db Da [ Db Da n Db Db n Da Set

L D fz 2 C W uaz D ubzg

bull For the convex lens domain ˝ab D Da Db the situation is similar to that ofa half-disk in Sect 76 the modified Schwarz potential is u D minua ub andthe unique mother body sits on ˝ab L the positive density with respect to thearc length measure there being the size of the jump of the normal derivative of uwhich is the same as jrub uaj (on ˝ L) Denote this mother body by ab

bull For the moon-shaped domain ˝ab D Da n Db the situation is somewhatsimilar to that of the lens domain the (presumably unique) mother body is apositive measure sitting on ˝ab L and the modified Schwarz potential still isminua ub where however it is now the restriction of ub to the exterior of Db

which comes inbull Similarly for ˝ba D Db n Dabull For˝aCb D Da [Db which is not convex the line segment Da DbL divides

the domain into two parts in such a way that the modified Schwarz potential uis represented by ua in the part containing the point a and by ub in the partcontaining b Within Da Db this gives u D maxua ub hence it is actually

112 7 Examples

Fig 710 Zeros of the orthogonal polynomials Pn n D 40 50 60 for ˝aCb with b D a D 1

opposite to the representation in the case of ˝ab The matching on Da Db L will now result in a negative contribution to the mother body more exactlythe contribution will be ab In addition there will be (positive) contributionsfrom the point masses at a and b In summary one mother body (which howeverviolates the positivity requirement iii in Chap 6) is

aCb D 1C a2ıa C 1C b2ıb ab

There are also two other mother bodies which fulfill all the requirements inChap 6 but which are decomposable in a non-desirable sense In the symmetriccase a C b D 0 they do not respect the symmetry of ˝aCb They are obtainedby simply adding the mother bodies of each piece in the decompositions (up tonull-sets) ˝aCb D Da [˝ba D Db [˝ab

The symmetric case with b D a D 1 is illustrated in Fig 710

78 External Disk

In general if˝ is the disjoint union of two domains say˝ D ˝1[˝2˝1˝2 D then E˝zw D E˝1zwE2zw and for example

H˝zw D H˝1zwE˝2zw for zw 2 ˝1

Thus H˝zw has H˝1zw as a main term inside ˝1 but is still is influencedby ˝2 via E˝2zw However the singularity structure inside ˝1 is completelyindependent of ˝2 By singularity structure we mean objects like mother bodies j

78 External Disk 113

for ˝j ( j D 1 2) or singularities for analytic continuations across the boundary ofthe exterior versions Fjzw of Ejzw

To make this precise consider the analytic extension into ˝1 of

F˝zw D F˝1zwF˝2zw

assuming that ˝1 is analytic Denoting by S1z the Schwarz function for ˝1 thatanalytic extension is given by

F˝zw D z S1wS1z Nwz wNz Nw E˝1zwF˝2zw zw 2 ˝1

Here E˝1zw is defined precisely as in (21) and in the last factor we couldalso have written E˝2zw in place of F˝2zw The mentioned singularities areexactly the singularities of the Schwarz function S1z equivalently of the Cauchytransform C˝1z inside ˝1 The point we wish to make is that S1z and C˝1zare local objects completely independent of ˝2 in contrast to what is the casefor the exponential transform and even worse for the global spaces objects suchas the Bergman orthogonal polynomials as we shall see below In other wordsmother bodies and related structures are local objects when considering disjointunion of domains This makes it believable that also asymptotic zero distributionsfor orthogonal polynomials should also local objects at least approximately Andthis is also what is seen in our numerical experiments below

If ˝ is the disjoint union of a non quadrature domain and a disk then the diskwill probably not be seen by the asymptotic distribution of zeros because only onezero is needed to ldquokillrdquo the disk Similarly for the disjoint union of a quadraturedomain and a non quadrature domain the quadrature domain will (probably) beinvisible in the limit

781 Numerical Experiment Ellipse Plus Disk

The disjoint union of an ellipse and a disk case is in sharp contrast with the Bergmanspace picture where different ldquoislandrdquo affect at distance the distribution of zeros ofthe respective complex orthogonal polynomials see [50] (Figs 711 and 712)

The external curve in the Bergman space picture is a singular level line of theGreenrsquos function of the union of the two domains In this case its Schwarz reflectionin the two boundaries is partially responsible for the root distribution For details onthe Bergman zero distribution see [50]

782 Numerical Experiment Pentagon Plus Disk

The numerical experiments supporting the following two images of a pentagon andan external disk illustrate again the sharp contrast between the zero distribution of

114 7 Examples

Fig 711 Zeros of the exponential orthogonal polynomials Pn n D 10 15 20 for the disjointunion of an ellipse and a disk

Fig 712 Zeros of the Bergman orthogonal polynomials Pn n D 80 90 100 for the disjointunion of an ellipse and a disk

Fig 713 Zeros of the exponential orthogonal polynomials Pn n D 40 60 80 for the disjointunion of a regular pentagon and a disk

complex orthogonal polynomials associated to the exponential transform respec-tively the Bergman space metric Like in the previous picture the external curvein the Bergman space picture is a singular level line of the Greenrsquos function of theexterior domain much responsible for the root distribution in the Bergman spacesituation (Figs 713 and 714)

79 Abelian Domains 115

Fig 714 Zeros of the Bergman orthogonal polynomials Pn n D 80 90 100 for the disjointunion of a regular pentagon and a disk

79 Abelian Domains

We consider here the case that ˝ is a quadrature in a slightly extended sense forexample that

h dA D cZ a

ah dx C

ckhak (712)

holds for all integrable analytic functions h in ˝ with a c ck gt 0 We assumethat ˝ is simply connected and then the conformal map from the unit disk willcontain logarithmic terms to account for the line segment in addition to the rationalterms needed for the point evaluations See in general [16] (Ch 14) [32 107] (˝is an Abelian domain in the sense of [107]) With emphasis on multiply connecteddomains systematic constructions (analytical and numerical) of quadrature domainsfor sources along curves are given in [73]

The simplest possible case is obtained by taking f W D ˝ of the form

f D A log1C ˛

1 ˛C B (713)

where 0 lt ˛ lt 1 AB gt 0 This gives

h dA D AZ a

ah dx C 2˛AB h0

where a D A log 1C˛21˛2 C B˛ gt 0 This example has the slight disadvantage that the

quadrature node z D 0 lies on the support of the line integral If one wants to avoid

116 7 Examples

that a next simplest example can be taken as

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

with 0 lt ˇ lt 1The computations are most easily performed by writing h as a derivative say

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

which is easily computed as a sum of residuesRecall that Corollary 54 says that under certain assumptions isolated quadra-

ture nodes such as the ak in (712) are ignored in the asymptotics of the zeros

710 Disjoint Union of a Hexagon and a Hypocycloid

In Fig 715 we have an example of a disjoint union of two non-quadrature domainsand it indicates to what extent the asymptotic zero distributions in the two domainsinfluence each other Only in the case of small separation there will be anysubstantial influence

Fig 715 Zeros of theorthogonal polynomials Pnn D 50 60 70 for a disjointunion of a hexagon and ahypocycloid

711 A Square with a Disk Removed 117

711 A Square with a Disk Removed

Choosing for example

˝ D fz 2 C W jxj lt 1 jyj lt 1 jzj gt Rg

where 0 lt R lt 1 the modified Schwarz potential u is obtained as the minimum offive such potentials one for each part of the boundary

uz D 1

2minfjx 1j2 jx C 1j2 jy 1j2 jy C 1j2 1

2jzj2 R2 log

jzj2R2

The (presumably unique) mother body for the density D 1 is then obtainedfrom (615)

The zeros for this doubly connected domain are illustrated in Fig 716

Fig 716 Zeros of the orthogonal polynomials Pn n D 40 60 90 for ˝ as in Sect 711 withR D 025

Chapter 8Comparison with Classical Function Spaces

Abstract Two classical quantizations of planar shapes via Toeplitz operatorsacting on Bergman space respectively Hardy-Smirnov space are briefly comparedto the new hyponormal quantization developed in these lecture notes

81 Bergman Space

It is possible and desirable for a comparison basis to create Hilbert spaces analogousto H ˝ or Ha˝ with other positive definite kernels in place of Hzw Onenatural choice is the Bergman kernel Kzw producing a Hilbert space K ˝

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Note that in the case of a quadrature domain ˝ the above sesquilinear form hasfinite rank We shall not pursue the analysis of this alternative scenario in this paperNote however that if one replaces in the above inner product the Bergman kernel byits complex conjugate then one recovers for analytic f and g the ordinary Bergmanspace inner product

Kzwf zgwdAzdAw D 1

f wgwdAw D h f gi2˝

Here the reproducing property

f w D 1

f zKzwdAz f 2 L2a˝ (81)

of the Bergman kernel was usedA further remark concerns the Friedrichs operator Let P W L2˝ L2a˝

denote the orthogonal projection onto the Bergman space ie the integral operator

given by the right member of (81) Then the linear transformation Pf D PNf is

120 8 Comparison with Classical Function Spaces

analogous to our previously studied operator H D ˇ ı ˛ see (37)

Pf w D 1

f zKzw dAz f 2 L2˝

In particular the range consists of anti-analytic functions Conjugating the resultgives the Friedrichs operator f 7 PNf which hence has the representation (onswitching names of the variables)

PNf z D 1

Kzwf w dAw

It maps L2a˝ into itself but has the disadvantage of being linear only over R (sosometimes one studies its square instead)

Finally we may remark on extension properties to compact Riemann surfaceKeeping w 2 ˝ fixed it is well-known that Kzwdz considered as a differential inz 2 ˝ extends to a meromorphic differential on the Schottky double O of˝ with adouble pole at the mirror point of w as the only singularity In other words Kzwextends to a meromorphic section of the canonical line bundle on the compactRiemann surface O As such it is uniquely determined by the natural periodicityconditions it satisfies (the integrals over those cycles in a canonical homology basiswhich are complementary to those cycles which lie entirely in the plane vanish)The analysis can be extended to both variables simultaneously so that Kzwdzd Nwbecomes a meromorphic double differential on O O See for example [96] for theabove matters

The above is to be compared with the corresponding properties of Hzwdiscussed in Sect 24 for fixed w 2 ˝ it extends as a holomorphic section of thehyperplane section bundle on the compact Riemann surface P (etc)

82 Faber Polynomials

Faber polynomials have a rich history spanning over more than a century Theirimprint on univalent function theory and polynomial approximation in the complexdomain is long lasting The aim of the present section is to draw a parallelbetween Faber polynomials and the orthogonal polynomials associated to theexponential transform Without aiming at full generality we confine ourselves toover conservative smoothness assumptions

Let ˝ C be a connected and simply connected bounded domain with realanalytic smooth boundary D ˝ Let

z D w D a1w C a0 C a1w

82 Faber Polynomials 121

be a conformal mapping of P n D onto P n˝ normalized so that 1 D 1 and 01 gt 0 We denote by

w D z D c1z C c0 C c1z

the inverse conformal map convergent in a neighborhood of z D 1 Faberrsquospolynomial of degree n is the unique polynomial fn satisfying at the level of formalLaurent series

fn w D wn C Rnw1

where Rnz is a power series without constant termCauchyrsquos formula proves the generating series definition

See for instance Ullman [106]We will show how to include this classical construction of polynomials attached

to a planar domain to our operator theory setting Specifically we consider theToeplitz operator with symbol

T h D P h h 2 H2T

Above H2 D H2T denotes the Hardy space of the unit disk with orthonormalbasis wk1kD0 and P W L2T d

2 H2 is the orthogonal projection often called the

Szegouml projection When analytically extending the functions from their boundaryvalues

Phz D 1

Proposition 81 Assume the Jordan curve is real analytic smooth Then theexterior conformal mapping analytically extends across the unit circle theToeplitz operator T has trace-class self-commutator and the following spectralpicture holds

T D ˝ essT D

with principal function g D ˝

Proof Let S D Tz denote the unilateral shift on Hardy space Since the function is smooth on the circle we can write

T D a1S C a0 C a1S C

where the series is convergent in operator norm Moreover due to the smoothnessof it is well known that in this case that the commutator ŒT

T is trace-classWriting

Q D a1S C a2S2 C

we note that Q is an analytic Toeplitz operator hence subnormal Consequently theself-commutator ŒQQ is non-positive This implies the bound

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

It is also well known that the essential spectrum of T is equal to the image of T by that is

Let 2 ˝ Then r curren 0 for all 2 T and r 1 Hence the Toeplitzoperator T r is Fredholm And so is its homothetic

r1T r D a1S C a0

But the latter can be deformed (r 1) in the space of Fredholm operators to a1Shence it has Fredholm index equal to 1 This proves that ˝ T and

indT D 1 2 ˝

If hellip ˝ then the function is invertible on T and has winding numberzero Then a factorization of the symbol into a product of an analyticinvertible function inside the disk and one analytic and invertible outside the diskshows that T is invertible

The infinite Toeplitz matrix associated to T in the basis formed by themonomials is

a0 a1 a2 a1 a0 a10 a1 a0

The cyclic subspaces

HnC1 D spanf1T 1 Tn 1g D spanf1w wng

form exactly the standard scale Let n denote the orthogonal projection of H2 ontoHn and set

T n D nT n

for the finite central truncation of the above Toeplitz matrixThe very definition of the Faber polynomial implies

Tfn 1 D wn n 0

On the other hand the inner product

Œ p q WD h pT 1 qT 1i

is non-degenerate on complex polynomials p q 2 CŒz hence we can speak aboutthe associated orthonormal polynomials Specifically there exists for every non-negative integer n a polynomial Fn of degree n such that

FnT 1 D wn n 0

We will call them quantized Faber polynomialsHowever the other natural inner product

f p qg D hTpı 1Tqı 1i D PV1

p eitq eis

1 eistdtds

has diagonal singularities comparable to the Hilbert space structure studied in theprevious chapters

As a matter of fact the Hessenberg matrix representing the multiplication by zin the basis fn1nD0 differs from the Toeplitz matrix associated to the inverse seriesz only by a rank-one perturbation More precisely adopting the normalizationa1 D c1 D 1 one knows that Faber polynomials fn are the characteristic polynomialsof the finite central truncations of the infinite matrix

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

see Eiermann and Varga [20] The distribution of zeros and asymptotics of thecounting measures of the Faber polynomials fn1nD0 is understood in detail dueto many recent contributions notably Ullman [106] and Kuijlaars-Saff [68]

The good news is that the zero asymptotics of the quantized Faber polynomialsis also well understood in the case of finite banded Toeplitz matrices that is forconformal mappings with finitely many negative modes

C C an

More exactly the cluster of eigenvalues of the finite central truncations T n aswell as the asymptotics of the counting measures is understood in this case in termsof and relation with classical potential theory We refer to the classical works ofSchmidt-Spitzer Hirschman and Ullman amply commented and carefully exposedin the monograph by Boumlttcher and Grudsky [11]

For our essay this case is quite relevant as the complement of the domain ˝ isa quadrature domain on the Riemann sphere

In order to illustrate what it is at stake in the asymptotic analysis of the zeros ofFaber polynomials we reproduce from [106] and [68] a few fundamental results Let

0 D lim supn1

jcnj1=n

denote the radius of convergence of the inverse conformal mapping The fibres ofthe map

W fz W jzj gt 0g C

play a crucial role First we isolate after Ullman the complement of the range of

C0 D fw 2 C W 1fwg D g

This is a compact subset of the complex plane because the range of covers aneighborhood of the point at infinity Second among the points in the range of we distinguish the set C1 of those w 2 C with the property that 1w has asingle element of largest modulus say and there the map is non-ramified that is0 curren 0 Since is non-ramified at z D 1 the set C1 covers a neighborhood ofinfinity

Ullman proved that all limit points of the Faber polynomials fk are contained inthe complement of C1 and that every boundary point of C1 is a limit point of thesezeros [106] Further on combining methods of potential theory with approximationtechniques of Toeplitz matrices Kuijlaars and Saff [68] refined the above pictureas follows Denote the counting measures of the zeros of Faber polynomials fk byk k 1

If C0 has no interior then the sequence k converges in weak-star topology toa positive measure whose support is equal to C1 If in addition C D C0 [ C1then is the equilibrium measure of C0 If int C0 is non-empty and connected thena subsequence of k will have the same limiting behavior It goes without sayingthat computing the sets C0 and C1 is quite challenging even for rational conformalmappings

Appendix AHyponormal Operators

We gather below some basic facts used throughout the text about hyponormaloperators By definition a linear bounded operator T acting on a Hilbert space His called hyponormal if the commutator inequality

ŒTT D TT TT 0

holds true in the operator sense That is for every vector x 2 H one has

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

Typical examples are subnormal operators and singular integral operators on theline with Cauchy type kernel Indeed if S D NjH is the restriction of a normaloperator to an invariant subspace H then

kSxk D kNxk D kNxk kPNxk D kSxk x 2 H

where P denotes the orthogonal projection of the larger Hilbert space onto HFor instance the multiplication Mz by the complex variable on the Bergman

space L2a˝ attached to a bounded planar domain˝ is subnormal hence hyponor-mal As for singular integral examples consider L2I dx where I is a closedinterval on the line Let a b 2 L1I with a D a ae Obviously the multiplicationoperator ŒXx D xx is self-adjoint on L2I dx The operator

ŒYx D axx bx

copy Springer International Publishing AG 2017B Gustafsson M Putinar Hyponormal Quantization of Planar DomainsLecture Notes in Mathematics 2199 DOI 101007978-3-319-65810-0

126 A Hyponormal Operators

is well defined as a principal value and bounded on L2 by the well known continuityof the Hilbert transform Then

ŒXYx D bx

ZIbyydy

hence T D X C iY is a hyponormal operator

ŒTT D 2iŒXY 0

It was this very specific example of singular integral transforms which boosted theearly discoveries related to hyponormal operators

Proposition A1 The spectral radius of a hyponormal operator is equal to its norm

Proof It is known that the spectral radius of T can be computed as jTjsp Dlimn kTnk1=n We will prove that for a hyponormal operator T one has

kTnk D kTkn n 1

Indeed let x 2 H and fix a positive integer n By assumption

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTnk2 D kTnTnk kTn1TTnk kTn1kkTTnk

kTn1kkTnC1k D kTn1kkTnC1k

If we suppose kTpk D kTkp p n as an induction hypothesis we obtain

kTknC1 kTnC1k

which implies kTknC1 D kTnC1kThus there are no non-trivial quasi-nilpotent hyponormal operators

Proposition A2 A linear fractional transformation preserves the hyponormality ofan operator

Proof Let T be a hyponormal operator and assume hellip T Then the operatorI T1 is bounded and

ŒI T1 I T1 D

I T1I T1ŒTTI T1I T1 0

An affine transformation obviously leaves hyponormal operators invariantAs a consequence of the spectral mapping theorem we infer

kI T1k D 1

dist T

This simple observation has a non-trivial consequence at the level of numericalrange

Proposition A3 Let T 2 L H be a hyponormal operator The closure of itsnumerical range coincides with the convex hull of its spectrum

WT D convT

Proof The inclusion convT WT holds in general Assume that there existsa point 2 WT which is not in the convex hull of the spectrum By an affinechange of variables we can assume that T is contained in a disk centered at zeroof radius r and jj gt r Since the spectral radius of T equals its norm we findkTk r and on the other hand

hTx xi D

for a unit vector x That is jj jhTx xij r a contradictionMore refined operations leave the class of hyponormal operators invariant For

instance take T D X C iY to be a hyponormal operator written in cartesian formand let S be a bounded operator in the commutant of X Then X C iSYS is alsohypo-normal

ŒX iSYSX C iSYS D 2iŒXSYS D S2iŒXYS D SŒTTS 0

In particular if S D EXı is the spectral projection associated to a Borel set ı Rone obtains a very useful cut-off operation of hyponormal operators One of the non-trivial implication is that in this case EXıTEXı is a hyponormal operate whosespectrum is localized to the band ı R C

In this respect it is worth recording a non-trivial spectral mapping projectionresult

Theorem A1 (Putnam [82]) Let T D XCiY be a hyponormal operator written incartesian form The projection of the spectrum T onto the x-axis coincides withthe spectrum of X

Hence the spectrum of the cut-off operator EXıTEXı lies in the vertical bandwith ı as base See [74] for proofs and more details

One can reverse the flow from abstract to concrete prove first that the real andimaginary parts of a pure hyponormal operators have absolute continuous spectrumand then build a general one variable singular integral functional model In thisprocess one needs precise estimates We recall two of them

A deep inequality related to non-negative self-commutators was discovered byPutnam It asserts that for every hyponormal operator T one has

kŒTTk Area T

Thus every hyponormal operator with ldquothinrdquo spectrum is normal in particular ahyponormal finite dimensional matrix is automatically normal

As a matter of fact not only the norm but even the trace of a hyponormal operatorT enters into a similar inequality This bears the name of Berger and Shaw and itstates

TraceŒTT mT

Area T

where mT stands for the rational multiplicity of T that is the minimal number ofvectors hj 1 j mT so that f Thj span the whole Hilbert space on which Tacts where f is an arbitrary rational function analytic in a neighborhood of T

Using these inequalities and cartesian cut-off operations one can prove that thespectrum of a pure hyponormal operators has positive planar density at every pointof it As a matter of fact every compact subset of the complex plane with this densityproperty is the spectrum of a hyponormal operator

The main concern in this lecture notes is the class of hyponormal operatorswith rank-one self commutator Their classification and spectral analysis is classicalby now thanks to fundamental contributions of Pincus Carey L Brown HeltonHowe Clancey D Xia The monograph [74] contains an overview of the theory ofhyponormal operators and a couple of chapters are devoted to this class of operatorsWe confine ourselves to a few identities used in the main body of the lecture notes

Let T 2 L H be a linear bounded operator acting on a separable infinitedimensional space and possessing rank-one self-commutator

ŒTT D ˝

We also assume that T is irreducible that is the linear span of vectors TnTmnm 0 is dense in H Let zw 2 C be points outside the spectrum of T Then theresolvents T w1 and T z1 exist and the multiplicative commutator

T zT wT z1T w1

is in the determinant class (that is the identity plus a trace-class operator) and

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

Mutatis mutantis we can perform the same operation on the multiplicativecommutator with reversed order of factors

detT wT zT w1T z1 D

detŒI C ˝ T w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

Since the product of the two commutators is the identity we infer

Œ1C hT z1 T w1iŒ1 hT w1 T z1i D 1 (A1)

The unitary equivalence orbit of the irreducible operator T subject to thecommutation relation ŒTT D ˝ is encoded into the germ at infinity of theanalytic-antianalytic determinantal function

Ezw D 1 hT w1 T z1i jzj jwj gt kTk

Hence also in the germ at infinity of the function

EzwD 1C hT z1 T w1i jzj jwj gt kTk

The main character of our study is the function E and its exponential representationas a double Cauchy transform

Theorem A2 (Pincus [76]) The integral representation

1 hT w1 T z1i D exp1

z w jzj jwj gt kTk

establishes a one-to-one correspondence between all irreducible hyponormal oper-ators T with rank-one self-commutator ŒTT D ˝ and L1-classes of Borelmeasurable functions g W C Œ0 1 of compact support

For the original proof see [76] The function g is called the principal functionof the operator T and it can be regarded as a generalized Fredholm index which isdefined even for points of the essential spectrum

A simple and far reaching application of the operator counterparts of the abovedeterminants is the ldquodiagonalizationrdquo in a concrete functional model of the operatorsT respectively T Specifically let denote a circle centered at z D 0 of radiusR gt kTk Then Riesz functional calculus yields for any couple of polynomialsf g 2 CŒz

h f T gTi D 1

f ugvdudv

Eu v (A2)

while in complete symmetry

hgT f Ti D 1

f ugvEu vdudv (A3)

To illustrate the nature of the above integral representation we outline a novelproof of a classical inequality due to Ahlfors and Beurling Specifically if anessentially bounded function f on C is non-negative and integrable with respectto the two dimensional Lebesgue measure then

f wd Areaw

w zj2 kf k1kf k1

for all z 2 CIndeed we can assume that f has compact support and 0 f 1 ae hence

it is equal to the principal function of a hyponormal operator T with rank-one self-commutator ŒTT D ˝ Then

h T z1i D 1

f wd Areaw

and on the other hand

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

A remarkable and more symmetric formula involving functional calculi wasdiscovered by Helton and Howe [58] We state it in our context of rank-one self-commutator although it is true for trace-class self-commutators

traceŒ pTT qTT D 1

J p qg dA p q 2 CŒz z (A4)

where J p q stands for the Jacobian of the two functions see [58 74] It was thisvery formula that was the source of Connesrsquo cyclic cohomology theory [15]

The principal function like the Fredholm index is functorial with respect toanalytic functional calculus Moreover it is invariant under Hilbert-Schmidt additiveperturbations and satisfies non-trivial bounds We reproduce below only one of theseinequalities For proofs and more comments see [74]

Theorem A3 (Berger [7]) Assume T S are hyponormal operators with trace-class self-commutator and let X be a trace-class operator with null kernel andcokernel If SX D XT then gS gT

In particular Bergerrsquos Theorem shows that the principal function of a cyclichyponormal operator is bounded by 1

In case the principal function g is the characteristic function of a bounded openset with real analytic smooth boundary the determinantal function also known asexponential transform Ezw analytically extends across the boundary as explainedin the main body of these notes This observation was used in the proof of a centralregularity result for free boundaries in two real dimensions cf Sect 64 in thesenotes

Among the early applications of the theory of the principal function anobservation of Basor and Helton [6] stands aside The two authors reproved andgeneralized the Szegouml limit theorem While today there are more refined variants ofthe theorem due to Widom and a few other authors the method used by Basor andHelton is relevant for the topics reviewed in Appendix We sketch below the mainidea

Let H2 D H2D denote Hardy space of the unit disk with orthogonal projectionP W L2T H2D A Toeplitz operator

T f D Pf f 2 H2

with continuous non-vanishing symbol 2 CT is known to be Fredholm withindex equal to negative of the winding number of along the unit circle Assumethat the symbol is smooth and has zero winding number Then the general theorybuilt around the Riemann-Hilbert problem yields a factorization

z D Czz z 2 T

where C is a smooth function on T which analytically extends to D while zis smooth and analytically extends to the complement of the disk and 1 D 1

It is easy to check for instance on monomials that

T D TTC

Denote by Pn the orthogonal projection of H2 onto the linear span of 1 z znThe finite central truncation Tn D PnTPn is a n C1 n C1 Toeplitz matrix andso are the triangular matrices

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

Pn is not equal to Tn exactly this discrepancy and theprincipal function formula proves Szegouml limit theorem Quite specifically let

G D exp1

logzdz

be the geometric mean of the invertible symbol The logarithm above is definedon C n 1 0 so that log1 D 0 Moreover

But the matrices TCn T

n are triangular with the identical entries equal to

RTCz dz

iz respectively 12

RTz dz

iz on the diagonal Hence

GnC1 D detTCn det T

Next linear algebra gives

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

Due to the smoothness assumption

det T1C

D det TTC

D detTT1

exists and is equal to the limit of the truncated determinants This observation plusHelton-Howe formula imply Szegouml-Widom limit Theorem

det Tn

GnC1 D detTT1 D exp1

JlogC logdA

Above J denotes the Jacobian of the two functions

Historical Notes

Quadrature domains were isolated in the work by Dov Aharonov and Harold SShapiro see in particular the seminal paper [1] Although originally motivated byunivalent function theory the central role of this class of semi-algebraic domainsfound unexpected reverberations in many areas of modern mathematics suchas fluid mechanics potential theory integrable systems free boundaries inverseproblems The collections of articles [19] offers a good glimpse on the status of thetheory of quadrature domains until 2005 A small selection of books survey articlesand more recent contributions are [16 26 46 70 89 101 107]

The idea of a potential theoretic skeleton goes right back to Isaac Newtonwho showed that the gravitational field produced by a homogenous ball coincidesoutside the ball by the field of a point mass located at the center of the ball Laterdevelopments involve for example a Preisschrift ldquoUumlber die analytische Fortsetzungdes Potentials ins Innere der anziehenden Massenrdquo by Herglotz available in [59](see also Chapter 1 in [101]) and in particular systematic work by DimiterZidarov and his collaborators on ldquoMaternal and most concentrated bodiesldquo (titleof Chapter III6 in [113]) The particular term ldquomother bodyrdquo was used informallyby O Kounchev for example and inspired by the ideas of this Bulgarian groupin geophysical potential theory an attempt to make the notion of a mother bodymathematically precise was made in [34] Some further developments can befound in [43 95 102] In recent years the ideas of a mother bodies relatedldquomadonna bodiesrdquo and other kinds of potential theoretic skeletons have turnedout to be relevant for asymptotic distributions of zeros of orthogonal polynomialseigenvalues of random matrices etc See for example [9 10 50]

Orthogonal polynomials have a long and glorious history spanning a centuryand a half Although originally streaming from real function theory questionssuch as extremal problems moment problems convergence of continued fractionseigenfunction expansions generating functions with a combinatorial flavor and soon complex variables turned out to be essential in studying the fine structure andasymptotic behavior of classical orthogonal polynomials The pioneering works of

136 Historical Notes

Fekete Szegouml Carleman Bergman Gonchar Suetin Widom (to name only a fewkey contributors to the subject) laid the foundations of complex analytic and poten-tial theoretic interpretation of the qualitative aspects of orthogonal polynomialsThis trend continues to bloom today with notable impact on quantum mechanicsstatistical mechanics and approximation theory The authoritative monographs[86 103] gave the tone for all later developments of the potential theoretic andcomplex function theoretic aspects of orthogonal polynomials In spite of the factthat the orthogonality studied in the present notes and in particular the class ofassociated orthogonal polynomials does not strictly fit in the framework of theseinfluential works there are too many similarities not to seek in future works aunifying explanation

One of the first uses of the exponential transform outside operator theorywas to prove regularity of certain free boundaries in two dimensions [36] Theboundaries in question were more precisely boundaries of quadrature domains (ina general sense) and those arising in the related area of Hele-Shaw flow movingboundary problems (Laplacian growth) See [51] for the latter topic The exponentialtransform provided in these cases a simplified route compared to that of Sakai [92ndash94] One advantage with the exponential transform is that it directly gives a realanalytic defining function of the free boundary On the other hand as a tool infree boundary theory it is far less flexible than the general methods developed byL Caffarelli and others within the framework of nonlinear PDE

In general the theory of free boundaries and their regularity has been a highlyactive and rapidly developing area for at least half a century The regularity theoryis in general very difficult in particular in higher dimensions A few standardreferences are [14 25 65 75]

The notion of hyponormal operator was introduced by Halmos under a differentname in the article [52] After a necessary period of piling up the naive foundationswell exposed in Halmos problem book [53] Putnam raised the subject to a higherspeed by discovering his famous inequality and by regarding everything from theperspective of commutator algebra [82] It was Joel D Pincus [76] who pushedforward hyponormal and semi-normal operators to a novel direction of quantumscattering theory by isolating a remarkable spectral invariant known as the principalfunction Afterwards it was clear that explicit models of hyponormal operatorsdepart from classical analytic functional spaces by involving singular integraltransforms with kernels of Cauchy type The trace formula for commutators ofHelton and Howe [58] put the principal function into a totally new focus withlinks to global analysis concepts such as curvature invariants characteristic classesand index formulas Out of this phenomenology Alain Connes [15] invented cycliccohomology an important chapter of modern operator algebras In this respect ouressay falls within low dimensional cyclic cohomology theory A synthesis of thetheory of hyponormal operators can be found in the monograph [74]

The exponential transform arose as the characteristic function of a Hilbertspace operator with trace-class self-commutator Functions of a complex variableencoding in an invariant way the unitary equivalence class of an operator are knownas characteristic functions they played a prominent role in the classification of

non-selfadjoint transformations streaming from the works of Moshe Livsic ondissipative operators and Nagy-Foias on contractive operators The lucid approachof Larry Brown [13] revealed the algebraic K-theoretic nature of the exponentialtransform as the determinant of a multiplicative commutator in an infinite dimen-sional setting On the other hand one can interpret the exponential transform inpurely function theoretic terms as a Riesz potential at critical exponent with clearbenefits for the multidimensional setting Or as a solution to a Riemann-Hilbertfactorization problem as in these notes very much in harmony with Pincusrsquo originalvision [76]

The Ritz-Galerkin method of converting an infinite dimensional and continuousoperator equation into a finite-dimensional discrete problem was widely usedand refined for more than a century in numerical analysis The more specificKrylov subspace approximation of a linear operator by its compressions to finitedimensional subspaces defined by the degree filtration of powers of the operatorsapplied to a cyclic vector is intimately linked to the theory of complex orthogonalpolynomials and has numerous supporters in the applied mathematics community[72] In particular the analysis and inversion of Toeplitz or Wiener-Hopf operatorsby their Krylov subspace compressions was thoroughly studied by several genera-tions of mathematicians see [11] for historical references and the intriguing (by itscomplexity and openness) picture of the field these days

Glossary

P D C [ f1g

DaR D fz 2 C W jz aj lt Rg D D D0 1

dA D dAz D dArea D dxdy

For˝ C a bounded open set

˝c D C n˝

˝e D C n˝ or P n˝ depending on context

j˝j D Area˝

f g2˝ D f gL2˝ D 1

f Ng dA Also k f kp˝ D jj f jjLp˝ (1 p 1)

L2a˝ Bergman space (analytic functions in L2˝)

DC Set of smooth test functions with compact support in C

OE Germs of functions holomorphic in an open set containing E C

Ezw The exponential transform of a bounded open set ˝ C Defined in allC C by (21)

Fzw The restriction of Ezw to ˝e ˝e and analytic continuations of thisSee (22)

Gzw A version of the exponential transform defined and analyticantianalyticin ˝ ˝e Gzw D Gw z See (23) (24)

Hzw The interior exponential transform defined and analyticantianalytic in˝ ˝ See (25)

C˝z C˝zw Cauchy transforms (ordinary and double) of ˝ Csee (28) (29)

140 Glossary

Cz Cauchy transform of a measure Written as Cf if d D f dA See Sect 61

U Logarithmic potential of a measure Written as U˝ if d D ˝ dA SeeSect 61

Mkj Mk Bkj D bkj Complex harmonic and exponential moments see Sect 22

Sz The Schwarz function of a real analytic curve See (220)

H ˝ A Hilbert space associated to the exponential transform see Sect 31

Ha˝ The subspace of H ˝ generated by analytic functions see (36)

h f gi Inner product in a Hilbert space in general

h f gi D h f giH ˝ Standard inner product in H ˝ (and Ha˝) See (34)

hh f gii D h Nf NgiH ˝ See (322)

h f giPXX The Pincus-Xia-Xia inner product in Ha˝ See (326)

f gduality D 12 i

f zgzdz f 2 O˝ g 2 O˝e0

H The operator L2˝ L2˝ with kernel Hzw defined by

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

G The operator with kernel Gzw defined by

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

Z The operator H ˝ H ˝ defined by Zf z D zf z

NZ The operator H ˝ H ˝ defined by NZf z D Nzf z

C The operator H ˝ H ˝ defined by Cf D Cf (Cauchy transform of f )

NC Related to C by NCf D CNf

L H The set of bounded linear operators on a Hilbert space H

C1H The set of those A 2 L H with jAj1 D trp

AA lt 1 (finite trace norm)

T Spectrum of an operator T 2 L H

WT D fhTx xi W x 2 H kxk D 1g the numerical range of an operator T 2L HMore special notations appear in the text

References

1 D Aharonov HS Shapiro Domains on which analytic functions satisfy quadratureidentities J Anal Math 30 39ndash73 (1976)

2 LV Ahlfors Complex Analysis An Introduction of the Theory of Analytic Functions of OneComplex Variable 2nd edn (McGraw-Hill New York 1966)

3 NI Akhiezer The Classical Moment Problem and Some Related Questions in Analysis(Hafner Publishing New York 1965) Translated by N Kemmer

4 H Alexander J Wermer Several Complex Variables and Banach Algebras Graduate Textsin Mathematics vol 35 3rd edn (Springer New York 1998)

5 M Andersson M Passare R Sigurdsson Complex Convexity and Analytic Functionals vol225 Progress in Mathematics (Birkhaumluser Basel 2004)

6 E Basor J William Helton A new proof of the Szego limit theorem and new results forToeplitz operators with discontinuous symbol J Operator Theory 3(1) 23ndash39 (1980)

7 CA Berger Intertwined operators and the Pincus principal function Integr Equ OperTheory 4(1) 1ndash9 (1981)

8 L Bers An approximation theorem J Anal Math 14 1ndash4 (1965)9 P Bleher G Silva The mother body phase transition in the normal matrix model (2016)

arXiv16010512410 R Boslash gvad B Shapiro On mother body measures with algebraic Cauchy transform Enseign

Math 62(1ndash2) 117ndash142 (2016)11 A Boumlttcher SM Grudsky Spectral Properties of Banded Toeplitz Matrices (Society for

Industrial and Applied Mathematics Philadelphia PA 2005)12 H Brezis AC Ponce Katorsquos inequality when u is a measure C R Math Acad Sci Paris

338(8) 599ndash604 (2004)13 LG Brown The Determinant Invariant for Operators with Trace Class Self Commutators

Lecture Notes in Mathematics vol 345 (Springer Berlin 1973) pp 210ndash22814 LA Caffarelli The obstacle problem revisited J Fourier Anal Appl 4(4ndash5) 383ndash402

(1998)15 A Connes Noncommutative Geometry (Academic San Diego CA 1994)16 PJ Davis The Schwarz Function and its Applications The Carus Mathematical Monographs

vol 17 (The Mathematical Association of America Buffalo NY 1974)17 P Diaconis W Fulton A growth model a game an algebra Lagrange inversion and

characteristic classes Rend Sem Mat Univ Politec Torino 49(1) 95ndash119 (1993) 1991Commutative algebra and algebraic geometry II (Italian) (Turin 1990)

18 WF Donoghue Jr Monotone Matrix Functions and Analytic Continuation (Springer NewYork Heidelberg 1974) Die Grundlehren der mathematischen Wissenschaften Band 207

142 References

19 P Ebenfelt B Gustafsson D Khavinson M Putinar Preface in Quadrature Domainsand Their Applications Operator Theory Advances and Applications vol 156 (BirkhaumluserBasel 2005) pp viindashx

20 M Eiermann RS Varga Zeros and local extreme points of Faber polynomials associatedwith hypocycloidal domains Electron Trans Numer Anal 1 49ndash71 (1993) (electronic only)

21 A Eremenko E Lundberg K Ramachandran Electrstatic skeletons (2013)arXiv13095483

22 HM Farkas I Kra Riemann Surfaces Graduate Texts in Mathematics vol 71 2nd edn(Springer New York 1992)

23 RP Feynman AR Hibbs Quantum Mechanics and Path Integrals emended edition (DoverMineola NY 2010) Emended and with a preface by Daniel F Styer

24 T Frankel The Geometry of Physics 3rd edn (Cambridge University Press Cambridge2012) An introduction

25 A Friedman Variational Principles and Free-Boundary Problems Pure and AppliedMathematics (Wiley New York 1982) A Wiley-Interscience Publication

26 SJ Gardiner T Sjoumldin Quadrature domains for harmonic functions Bull Lond Math Soc39(4) 586ndash590 (2007)

27 SJ Gardiner T Sjoumldin Partial balayage and the exterior inverse problem of potential theoryin Potential Theory and Stochastics in Albac Theta Series in Advanced Mathematics (ThetaBucharest 2009) pp 111ndash123

28 SJ Gardiner T Sjoumldin Two-phase quadrature domains J Anal Math 116 335ndash354 (2012)29 P Griffiths J Harris Principles of Algebraic Geometry (Wiley-Interscience New York

1978) Pure and Applied Mathematics30 A Grothendieck Sur certains espaces de fonctions holomorphes I J Reine Angew Math

192 35ndash64 (1953)31 M Guidry Gauge Field Theories A Wiley-Interscience Publication (Wiley New York

1991) An introduction with applications32 B Gustafsson Quadrature identities and the Schottky double Acta Appl Math 1(3) 209ndash

240 (1983)33 B Gustafsson Singular and special points on quadrature domains from an algebraic

geometric point of view J Anal Math 51 91ndash117 (1988)34 B Gustafsson On mother bodies of convex polyhedra SIAM J Math Anal 29(5) 1106ndash

1117 (1998) (electronic)35 B Gustafsson Lectures on balayage in Clifford Algebras and Potential Theory Univ

Joensuu Dept Math Rep Ser vol 7 (University of Joensuu Joensuu 2004) pp 17ndash6336 B Gustafsson M Putinar An exponential transform and regularity of free boundaries in two

dimensions Ann Scuola Norm Sup Pisa Cl Sci (4) 26(3) 507ndash543 (1998)37 B Gustafsson M Putinar Linear analysis of quadrature domains II Isr J Math 119

187ndash216 (2000)38 B Gustafsson M Putinar Analytic continuation of Cauchy and exponential transforms

in Analytic Extension Formulas and Their Applications (Fukuoka 1999Kyoto 2000)International Society for Analysis Applications and Computation vol 9 (Kluwer Dordrecht2001) pp 47ndash57

39 B Gustafsson M Putinar Linear analysis of quadrature domains IV in QuadratureDomains and Their Applications Operator Theory Advances and Applications vol 156(Birkhaumluser Basel 2005) pp 173ndash194

40 B Gustafsson M Putinar Selected topics on quadrature domains Phys D 235(1ndash2) 90ndash100(2007)

41 B Gustafsson J Roos Partial balayage on riemannian manifolds (2016) arXiv16050310242 B Gustafsson M Sakai Properties of some balayage operators with applications to

quadrature domains and moving boundary problems Nonlinear Anal 22(10) 1221ndash1245(1994)

43 B Gustafsson M Sakai On potential-theoretic skeletons of polyhedra Geom Dedicata76(1) 1ndash30 (1999)

References 143

44 B Gustafsson M Sakai Sharp estimates of the curvature of some free boundaries in twodimensions Ann Acad Sci Fenn Math 28(1) 123ndash142 (2003)

45 B Gustafsson M Sakai On the curvature of the free boundary for the obstacle problem intwo dimensions Monatsh Math 142(1ndash2) 1ndash5 (2004)

46 B Gustafsson HS Shapiro What is a quadrature domain in Quadrature Domains andTheir Applications Operator Theory Advances and Applications vol 156 (Birkhaumluser Basel2005) pp 1ndash25

47 B Gustafsson VG Tkachev On the exponential transform of multi-sheeted algebraicdomains Comput Methods Funct Theory 11(2) 591ndash615 (2011)

48 B Gustafsson M Sakai HS Shapiro On domains in which harmonic functions satisfygeneralized mean value properties Potential Anal 7(1) 467ndash484 (1997)

49 B Gustafsson C He P Milanfar M Putinar Reconstructing planar domains from theirmoments Inverse Prob 16(4) 1053ndash1070 (2000)

50 B Gustafsson M Putinar EB Saff N Stylianopoulos Bergman polynomials on anarchipelago estimates zeros and shape reconstruction Adv Math 222(4) 1405ndash1460 (2009)

51 B Gustafsson R Teoderscu A Vasilrsquoevrsquo Classical and Stochastic Laplacian GrowthAdvances in Mathematical Fluid Mechanics (Birkhaumluser Basel 2014)

52 PR Halmos Normal dilations and extensions of operators Summa Brasil Math 2 125ndash134(1950)

53 PR Halmos A Hilbert Space Problem Book (D Van Nostrand Princeton NJ TorontoOntorio London 1967)

54 FR Harvey H Blaine Lawson Jr On boundaries of complex analytic varieties I AnnMath (2) 102(2) 223ndash290 (1975)

55 MX He EB Saff The zeros of Faber polynomials for an m-cusped hypocycloid J ApproxTheory 78(3) 410ndash432 (1994)

56 LI Hedberg Approximation in the mean by solutions of elliptic equations Duke Math J40 9ndash16 (1973)

57 H Hedenmalm N Makarov Coulomb gas ensembles and Laplacian growth Proc LondMath Soc (3) 106(4) 859ndash907 (2013)

58 JW Helton RE Howe Traces of commutators of integral operators Acta Math 135(3ndash4)271ndash305 (1975)

59 G Herglotz Gesammelte Schriften (Vandenhoeck amp Ruprecht Goumlttingen 1979) Withintroductory articles by Peter Bergmann S S Chern Ronald B Guenther Claus MuumlllerTheodor Schneider and H Wittich Edited and with a foreword by Hans Schwerdtfeger

60 L Houmlrmander The Analysis of Linear Partial Differential Operators I Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] vol256 (Springer Berlin 1983) Distribution theory and Fourier analysis

61 L Houmlrmander Notions of Convexity Progress in Mathematics vol 127 (Birkhaumluser BostonMA 1994)

62 K Huang Quantum Field Theory A Wiley-Interscience Publication (Wiley New York1998) From operators to path integrals

63 L Karp AS Margulis Newtonian potential theory for unbounded sources and applicationsto free boundary problems J Anal Math 70 1ndash63 (1996)

64 D Khavinson E Lundberg A tale of ellipsoids in potential theory Not Am Math Soc61(2) 148ndash156 (2014)

65 D Kinderlehrer G Stampacchia An Introduction to Variational Inequalities and TheirApplications Pure and Applied Mathematics vol 88 (Academic [Harcourt Brace JovanovichPublishers] New York-London 1980)

66 G Koumlthe Dualitaumlt in der Funktionentheorie J Reine Angew Math 191 30ndash49 (1953)67 MG Kreın AA Nudelrsquoman The Markov Moment Problem and Extremal Problems

(American Mathematical Society Providence RI 1977) Ideas and problems of PL Cebyševand AA Markov and their further development Translated from the Russian by D LouvishTranslations of Mathematical Monographs vol 50

144 References

68 ABJ Kuijlaars EB Saff Asymptotic distribution of the zeros of Faber polynomials MathProc Camb Philos Soc 118(3) 437ndash447 (1995)

69 JC Langer DA Singer Foci and foliations of real algebraic curves Milan J Math 75225ndash271 (2007)

70 S-Y Lee NG Makarov Topology of quadrature domains J Am Math Soc 29(2) 333ndash369(2016)

71 L Levine Y Peres Scaling limits for internal aggregation models with multiple sources JAnal Math 111 151ndash219 (2010)

72 J Liesen Z Strakoˇ s Krylov Subspace Methods Numerical Mathematics and ScientificComputation (Oxford University Press Oxford 2013) Principles and analysis

73 JS Marshall On the construction of multiply connected arc integral quadrature domainsComput Methods Funct Theory 14(1) 107ndash138 (2014)

74 M Martin M Putinar Lectures on Hyponormal Operators Operator Theory Advances andApplications vol 39 (Birkhaumluser Basel 1989)

75 A Petrosyan H Shahgholian N Uraltseva Regularity of Free Boundaries in Obstacle-Type Problems Graduate Studies in Mathematics vol 136 (American Mathematical SocietyProvidence RI 2012)

76 JD Pincus Commutators and systems of singular integral equations I Acta Math 121219ndash249 (1968)

77 JD Pincus D Xia JB Xia The analytic model of a hyponormal operator with rank oneself-commutator Integr Equ Oper Theory 7(4) 516ndash535 (1984)

78 J Polchinski String Theory Vol I Cambridge Monographs on Mathematical Physics(Cambridge University Press Cambridge 1998) An introduction to the bosonic string

79 M Putinar On a class of finitely determined planar domains Math Res Lett 1(3) 389ndash398(1994)

80 M Putinar Linear analysis of quadrature domains Ark Mat 33(2) 357ndash376 (1995)81 M Putinar Extremal solutions of the two-dimensional L-problem of moments J Funct

Anal 136(2) 331ndash364 (1996)82 CR Putnam Commutation Properties of Hilbert Space operators and Related Topics

Ergebnisse der Mathematik und ihrer Grenzgebiete Band 36 (Springer New York 1967)83 F Riesz BSz-Nagy Leccedilons drsquoanalyse fonctionnelle (Acadeacutemie des Sciences de Hongrie

Akadeacutemiai Kiadoacute Budapest 1952)84 C Rovelli Quantum Gravity Cambridge Monographs on Mathematical Physics (Cambridge

University Press Cambridge 2004) With a foreword by James Bjorken85 EB Saff NS Stylianopoulos Asymptotics for polynomial zeros beware of predictions

from plots Comput Methods Funct Theory 8(1ndash2) 385ndash407 (2008)86 EB Saff V Totik Logarithmic Potentials with External Fields Grundlehren der Math-

ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] vol 316(Springer Berlin 1997) Appendix B by Thomas Bloom

87 M Sakai A moment problem on Jordan domains Proc Am Math Soc 70(1) 35ndash38 (1978)88 M Sakai Null quadrature domains J Anal Math 40 144ndash154 (1982) 198189 M Sakai Quadrature Domains Lecture Notes in Mathematics vol 934 (Springer Berlin

1982)90 M Sakai Applications of variational inequalities to the existence theorem on quadrature

domains Trans Am Math Soc 276(1) 267ndash279 (1983)91 M Sakai Solutions to the obstacle problem as Green potentials J Anal Math 44 97ndash116

(19841985)92 M Sakai Regularity of a boundary having a Schwarz function Acta Math 166(3ndash4) 263ndash

297 (1991)93 M Sakai Regularity of boundaries of quadrature domains in two dimensions SIAM J Math

Anal 24(2) 341ndash364 (1993)94 M Sakai Regularity of free boundaries in two dimensions Ann Scuola Norm Sup Pisa Cl

Sci (4) 20(3) 323ndash339 (1993)

References 145

95 TV Savina BYu Sternin VE Shatalov On a minimal element for a family of bodiesproducing the same external gravitational field Appl Anal 84(7) 649ndash668 (2005)

96 M Schiffer DC Spencer Functionals of Finite Riemann Surfaces (Princeton UniversityPress Princeton NJ 1954)

97 MD Schwartz Quantum Field Theory and the Standard Model (Cambridge UniversityPress Cambridge 2014)

98 J Sebastiatildeo e Silva Analytic functions and functional analysis Port Math 9 1ndash130 (1950)99 H Shahgholian Unbounded quadrature domains in Rn n 3 J Anal Math 56 281ndash291

(1991)100 HS Shapiro Unbounded quadrature domains in Complex Analysis I (College Park MD

1985ndash86) Lecture Notes in Mathematics vol 1275 (Springer Berlin 1987) pp 287ndash331101 HS Shapiro The Schwarz Function and its Generalization to Higher Dimensions University

of Arkansas Lecture Notes in the Mathematical Sciences vol 9 (Wiley New York 1992) AWiley-Interscience Publication

102 T Sjoumldin Mother bodies of algebraic domains in the complex plane Complex Var EllipticEqu 51(4) 357ndash369 (2006)

103 H Stahl V Totik General orthogonal polynomials volume 43 of Encyclopedia ofMathematics and its Applications Cambridge University Press Cambridge 1992

104 MH Stone Linear Transformations in Hilbert Space American Mathematical SocietyColloquium Publications vol 15 (American Mathematical Society Providence RI 1990)Reprint of the 1932 original

105 K Strebel Quadratic Differentials Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[Results in Mathematics and Related Areas (3)] vol 5 (Springer Berlin 1984)

106 JL Ullman Studies in Faber polynomials I Trans Am Math Soc 94 515ndash528 (1960)107 AN Varchenko PI Etingof Why the Boundary of a Round Drop Becomes a Curve of Order

Four AMS University Lecture Series 3rd edn (American Mathematical Society ProvidenceRI 1992)

108 JL Walsh Interpolation and Approximation by Rational Functions in the Complex Domain4th edn American Mathematical Society Colloquium Publications vol XX (AmericanMathematical Society Providence RI 1965)

109 RO Wells Jr Differential Analysis on Complex Manifolds Prentice-Hall Series in ModernAnalysis (Prentice-Hall Englewood Cliffs NJ 1973)

110 H Widom Extremal polynomials associated with a system of curves in the complex planeAdv Math 3 127ndash232 (1969)

111 D Xia Analytic Theory of Subnormal Operators (World Scientific Hackensack NJ 2015)112 DV Yakubovich Real separated algebraic curves quadrature domains Ahlfors type

functions and operator theory J Funct Anal 236(1) 25ndash58 (2006)113 D Zidarov Inverse Gravimetric Problem in Geoprospecting and Geodesy Developments in

Solid Earth Geophysics vol 19 (Elsevier Amsterdam 1990)

algebraic domain 40analytic continuation 86analytic functional 43analytic model 33 34analytic moment 11analyticity of free boundary 86annihilation and creation

operators 36annulus 94

Bergman inner product 33Bergman kernel 119Bergman space 119

Caley-Hamilton theorem 58canonical line bundle 120carrier (of analytic functional) 43Cauchy kernel 32Cauchy transform 8Cauchy transform (as an operator) 27characteristic polynomial 58Chebyshev polynomial 98Chern class 17Cholesky decomposition 72cohyponormal operator 28 31complex moment 11convex carrier 91counting measure 47 57cross ratio 10

defect operator 70determinant of an operator 29double Cauchy transform 8dual basis 38 50

electrostatic skeleton 80ellipse 63 97equilibrium measure 104exponential moment 11exponential transform 7

Faber polynomial 120Feynman integral 36finite central truncation 57finitely determined domain 40four variable exponential transform 10Fredhom index 130Friedrichs operator 119

generalized lemniscate 56genus formula 100Gram matrix 26Greenrsquos function 48 105 113

Hardy space 35harmonic moment 11Hessenberg matrix 52 123hyperfunction 43hyperplane section bundle 17hypocycloid 99hyponormal operator 31 125

Jacobi matrix 98Jacobi-Toeplitz matrix 53

lemniscate 103line bundle 16logarithmic potential 78

148 Index

madonna body 80 105minimal carrier 77 82minimal convex carrier 43minimal polynomial 58modified Schwarz potential 89 108 117mother body 44 78

numerical range 58 70 127

order of a quadrature domain 41orthogonal polynomial 47

Padeacute approximation 60partial balayage 79plurisubharmonic function 51positive definite matrix 13preface viiprincipal function 31 130

quadratic differential 90quadrature domain 40quadrature domain for subharmonic functions

80quadrature domain in the wide sense 44quantized Faber polynomial 123

rational multiplicity 128real central truncation 74reproducing kernel 15ridge 105Riemann-Hilbert problem 18Riemann-Hurwitz formula 96 100

Schottky double 96 120Schurrsquos theorem 13Schwarz function 16Schwarz reflection 113shift operator 27Silva-Koumlthe-Grothendieck duality 37skeleton 78spectral radius 126symbol 121Szegouml limit theorem 131

three term relation 63Toeplitz matrix 122Toeplitz operator 121trace norm 58trace-class operator 29trace-class perturbation 58

unilateral shift 54 122

LECTURE NOTES IN MATHEMATICS 123Editors in Chief J-M Morel B Teissier

Editorial Policy

1 Lecture Notes aim to report new developments in all areas of mathematics and theirapplications ndash quickly informally and at a high level Mathematical texts analysing newdevelopments in modelling and numerical simulation are welcome

Manuscripts should be reasonably self-contained and rounded off Thus they may andoften will present not only results of the author but also related work by other people Theymay be based on specialised lecture courses Furthermore the manuscripts should providesufficient motivation examples and applications This clearly distinguishes Lecture Notesfrom journal articles or technical reports which normally are very concise Articlesintended for a journal but too long to be accepted by most journals usually do not havethis ldquolecture notesrdquo character For similar reasons it is unusual for doctoral theses to beaccepted for the Lecture Notes series though habilitation theses may be appropriate

2 Besides monographs multi-author manuscripts resulting from SUMMER SCHOOLS orsimilar INTENSIVE COURSES are welcome provided their objective was held to presentan active mathematical topic to an audience at the beginning or intermediate graduate level(a list of participants should be provided)

The resulting manuscript should not be just a collection of course notes but should requireadvance planning and coordination among the main lecturers The subject matter shoulddictate the structure of the book This structure should be motivated and explained ina scientific introduction and the notation references index and formulation of resultsshould be if possible unified by the editors Each contribution should have an abstractand an introduction referring to the other contributions In other words more preparatorywork must go into a multi-authored volume than simply assembling a disparate collectionof papers communicated at the event

3 Manuscripts should be submitted either online at wwweditorialmanagercomlnm toSpringerrsquos mathematics editorial in Heidelberg or electronically to one of the series edi-tors Authors should be aware that incomplete or insufficiently close-to-final manuscriptsalmost always result in longer refereeing times and nevertheless unclear refereesrsquo rec-ommendations making further refereeing of a final draft necessary The strict minimumamount of material that will be considered should include a detailed outline describingthe planned contents of each chapter a bibliography and several sample chapters Parallelsubmission of a manuscript to another publisher while under consideration for LNM is notacceptable and can lead to rejection

4 In general monographs will be sent out to at least 2 external referees for evaluation

A final decision to publish can be made only on the basis of the complete manuscripthowever a refereeing process leading to a preliminary decision can be based on a pre-finalor incomplete manuscript

Volume Editors of multi-author works are expected to arrange for the refereeing to theusual scientific standards of the individual contributions If the resulting reports can be

forwarded to the LNM Editorial Board this is very helpful If no reports are forwardedor if other questions remain unclear in respect of homogeneity etc the series editors maywish to consult external referees for an overall evaluation of the volume

5 Manuscripts should in general be submitted in English Final manuscripts should containat least 100 pages of mathematical text and should always include

ndash a table of contentsndash an informative introduction with adequate motivation and perhaps some historical

remarks it should be accessible to a reader not intimately familiar with the topictreated

ndash a subject index as a rule this is genuinely helpful for the readerndash For evaluation purposes manuscripts should be submitted as pdf files

6 Careful preparation of the manuscripts will help keep production time short besidesensuring satisfactory appearance of the finished book in print and online After accep-tance of the manuscript authors will be asked to prepare the final LaTeX source files(see LaTeX templates online httpswwwspringercomgbauthors-editorsbook-authors-editorsmanuscriptpreparation5636) plus the corresponding pdf- or zipped ps-file TheLaTeX source files are essential for producing the full-text online version of the booksee httplinkspringercombookseries304 for the existing online volumes of LNM) Thetechnical production of a Lecture Notes volume takes approximately 12 weeks Additionalinstructions if necessary are available on request from lnmspringercom

7 Authors receive a total of 30 free copies of their volume and free access to their book onSpringerLink but no royalties They are entitled to a discount of 333 on the price ofSpringer books purchased for their personal use if ordering directly from Springer

8 Commitment to publish is made by a Publishing Agreement contributing authors ofmultiauthor books are requested to sign a Consent to Publish form Springer-Verlagregisters the copyright for each volume Authors are free to reuse material contained intheir LNM volumes in later publications a brief written (or e-mail) request for formalpermission is sufficient

AddressesProfessor Jean-Michel Morel CMLA Eacutecole Normale Supeacuterieure de Cachan FranceE-mail moreljeanmichelgmailcom

Professor Bernard Teissier Equipe Geacuteomeacutetrie et DynamiqueInstitut de Matheacutematiques de Jussieu ndash Paris Rive Gauche Paris FranceE-mail bernardteissierimj-prgfr

Springer Ute McCrory Mathematics Heidelberg GermanyE-mail lnmspringercom

Preface

Contents

1 Introduction

2 The Exponential Transform

22 Moments
3 Hilbert Space Factorization
33 Linear Operators on H(Ω)
36 The Analytic Subspace Ha(Ω)
4 Exponential Orthogonal Polynomials
5 Finite Central Truncations of Linear Operators
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
8 Comparison with Classical Function Spaces
81 Bergman Space
A Hyponormal Operators
Historical Notes
Glossary
References
Index

Editors-in-ChiefJean-Michel Morel CachanBernard Teissier Paris

Advisory BoardMichel Brion GrenobleCamillo De Lellis ZurichAlessio Figalli ZurichDavar Khoshnevisan Salt Lake CityIoannis Kontoyiannis AthensGaacutebor Lugosi BarcelonaMark Podolskij AarhusSylvia Serfaty New YorkAnna Wienhard Heidelberg

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

Preface

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

viii Preface

Contents

1 Introduction 1

x Contents

Glossary 139

References 141

Index 147

Mk` DZC

2 1 Introduction

ŒTT D ˝

detT zT wT z1T w1 D

1 hT w1 T z1i D

expŒ 1

1 Introduction 3

traceŒpTT qTT D 1

Egzw D expŒ 1

z N Nw

4 1 Introduction

E˝zw D Qzw

E˝zw D 1 hTn w1 T

n z1i C Rnzw

pnzpnwn1XjD0

qjzqjw

1 Introduction 5

Egzw D exp Œ 1

z N Nw (21)

Fzw D Ezw z 2 ˝e w 2 ˝e (22)

Gzw D Ezw

Nz D Ezw

Nz Nw z 2 ˝ w 2 ˝e (23)

Gzw D Ezw

wD Ezw

z w z 2 ˝e w 2 ˝ (24)

Hzw D 2Ezw

NzwD Ezw

z wNz Nw z 2 ˝ w 2 ˝ (25)

Ezw D 1 C˝z

C˝z D 1

z^ d N (28)

C˝zw D 1

z^ d N

N Nw (29)

Cgz D 1

2C˝zw

and similarly

Nzw1 E˝zw D

(H˝zw zw 2 ˝ ˝0 zw hellip ˝ ˝ (211)

1 Ezw D 1

Hu vdAu

Nv Nw zw 2 C (212)

exp Œ 1

log j wj d

(Fzw zw 2 ˝e

Hzw zw 2 ˝

Re C˝zw D 1

Fzw D 1 R2

Gzw D 1

Nw Na zw 2 DaR DaRe

Gzw D 1

z a zw 2 DaRe DaR

Hzw D 1

H˝zw D 1

a ^ d N

N Nw d NN Nb

E˝z bE˝aw (215)

C˝zwI a b D 1

a ^ d N

N Nw d NN Nb

z ba w

22 Moments 11

f 0df f z

f 0df f a

22 Moments

Mkj D 1

Mk D Mk0 D 1

zkC1 NwjC1 (217)

B D 1 expM

B DXkj0

zkC1 NwjC1 M DXkj0

zkC1 NwjC1

k C 1Mkj Bkj DXpq

1 E˝zw DXkj0

zkC1 NwjC1

C˝zw DXkj0

zkC1 NwjC1

C˝z DXk0

zkC1 DXk0

E˝zk zjI ak aj 0

Proof We have

akj2 dA 0

E˝zk zjI ak ajDXkj

R2 z Nw D1X

zk Nwk

R2kC2 jzj jwj lt R

˛jızj X

ˇkıwk i DXjk

˛jKzjwk Nk

˛jızj 7X

˛jKzj

˛jKzj X

ˇkKwk iRK DXjk

˛jKzjwk Nk

h˚Kw iRK D hX

˛jKzj Kw iRK

˛jKzjw D ˚w

HD0Rzw H˝zw D 1

ED0Rn˝zw

b (219)

H˝zw D E˝zw

jz wj2 E˝b b

Nw NaNw Nb H˝zw

ˇ2 E˝z bE˝bw

E˝b b

Hf ˝ f z f w

HD f z f wD 1

EDnf ˝ f z f w

Sz D Nz z 2 ˝ (220)

ChernS Nw D 1

d logSz Nw D 1

d logNz Nw D 0

Gzw D 1

zC w C˝w

z2C Ojzj3 (223)

Gzwz w D Fzw w 2 ˝ z 2 ˝e (227)

Hzwz w D Gzw w 2 ˝ z 2 ˝ (228)

F1w D Fz1 D 1 (229)

C Ojzj2 (230)

Hzwz wd Nw DZ˝

Gzwd Nw D 2i z 2 ˝

HzwdAw D 1 z 2 ˝ (231)

Nz NbNz Nw

Na NwNa Nb

b ab EzwI a b

D(1 2 ˝0 hellip ˝

(f z z 2 ˝0 z 2 ˝e

Example 22 With

f z D Hzw

gz D NwHzw

hz D Gzw

CHwz D Gzw z 2 ˝e

Gzw D 1

Hz vdAv

Nv Nw z 2 ˝e w 2 ˝ (235)

f z D Gzw

gz D 1C NwGzw

hz D 1 Fzw

it follows that

C˝z D 1

Hu vdAu

u zdAv z 2 C (236)

C˝z D 1

support in C by

h f gi D 1

Hzwf zgwdAzdAw (34)

˛ W L1˝ H ˝ (36)

˛ W L2˝ H ˝

˛0 W H ˝0 L2˝0

ˇ W H ˝ L2˝

H D ˇ ı ˛ W L2˝ L2˝

Hf w D 1

Hzwf zdAz w 2 ˝ f 2 L2˝ (37)

h˛ f ˛giH ˝ D f HgL2˝ f g 2 L2˝ (38)

1 Ezw D 1

Hu vuk Nvj

zkC1 NwjC1 dAudAv

hzk zjiH ˝

zkC1 NwjC1

1 Ezw D 1 expŒ 1

z N Nw

D 1 expŒXkj0

zk zjL2˝

zkC1 NwjC1

Bkj D hzk zjiH ˝

hh 1iH ˝ D 1

hdA h 2 H ˝ (39)

Z W H ˝ H ˝ Zf z D zf z (310)

Hzwf zf wdAzdAw

pHa a jjZf jj D a2

Z D clos˝ (311)

Zf z D Nzf z (312)

Cgz D Cgz D 1

z z 2 ˝ (313)

1 ˝ 1 W h 7 hh 1i1

Z D Z C C (314)

ŒZC D 1 ˝ 1

ŒZZ D 1 ˝ 1 (315)

Hzwz wf zgwdAzdAw

Hzwz wf z

wCgwdAzdAw

Hzwf zCgwdAzdAw

This says that

ŒZC f z D z 1

1 E˝zw D hZ z11 Z w11i (317)

C˝z D hZ z11 1i (318)

kz D Z z11 (319)

Bkj D hZk1Zj1i

hh f gii D 1

Cf z D 1

f dAN Nz z 2 ˝

T D Z C C

ŒTT D 1 ˝ 1

C˝z D hh1T Nz11ii

Bkj D hhTj1Tk1ii

ŒAA D ˝

ŒTT D ˝

Bkj D hAkAji

respectively

Bkj D hTjTk

h f gi D 1

f Z D 1

If zZ z1 dz

same form as (325)

f gL2˝ D 1

h f giHa˝ D 1

h f giPXX D 1

zF D G D G 2 ˝ (330)

inner product as

Ezwdzdw

Fzwdzdw D 1

f wgwd Nw

k f Tk2 D 1

j f j2 jdjjG j C 1

f zf w

Fzwdzdw

ŒAA D1X

h f tf j i tii DZ

DŒ˚eiSŒ˚ (332)

˚ D log z 2 ˝

Gf w D 1

f gduality D 1

f zgzdz f 2 O˝ g 2 O˝e0 (334)

Gf w D 1

dEzwf zdz D 1

Fzwf zdz

h f giHa˝ D 1

Gf wgwd Nw D 1

f zGgzdz

G1w D C˝w w 2 ˝e

H1 D 1

qn D Gpn (336)

pkzqjzdz D ıkj

kz D 1

z 2 ˝

zqnd D qnz

kz D1X

nD0qnz pn

pnqnz 2 ˝ (337)

1 Ezw D h 1

qnzqnw (338)

j˝j 1

C˝z D h 1

z 1i D

q0z (339)

space itself isP

pnpnz z 2 ˝ (340)

h dA DmX

nk1XjD0

ckjhjak (341)

Sz D Nz for z 2 ˝ (343)

E˝zw D Qz NwPzPw

detBkj0kjd D 0

˛jO˝ W O˝ H ˝

dimHa˝ D d

0 D h f z

z w 1i D 1

f zdAz

z w w hellip ˝

h f giHa˝ DX

jhj c sup

jhj h 2 OD (346)

Cz D 1kz z 2 Dc

h 13i D 1

2z ˝ N13wHzw 13 2 O 0˝

˛ W O 0˝ H ˝

h 7 hh 1i D 1

h dA h 2 OD (347)

h dA DZ

h d h 2 OD (348)

h dA DZ

13n D 1

ızj (43)

nD0cnpn

cn D h f pni

lim supn1

jh f pnij1=n 1

lim supn1

jj f QnjjL1˝1=n 1

kz D 1

qnzpn (45)

qnz D h 1

z pni (46)

qnz D 1X

hk pnizkC1 D 1

nznC1 C O1

znC2 (47)

zD pn pnz

pnzqnz D h 1

zpn pni

Rkz D expŒg˝ez1

1 Ezw Dd1XnD0

qnzqnw

1 Ezw Dd1XkD0

Pw (48)

for n d and

qnz D Qdn1zPz

for 0 n lt d

Ina1 an D jjnY

kD1z akjj2

kD1z ak

akNajIna1 an D h

QniD1z ai

from which

nXkjD1

akNajIna1 ank

iD1z ai

nYiD1z ai

z aji 0

Vn conv ˝ (411)

jjPnzjj lt jj z a

z bPnzjj

norm it shows that

jjPnzjj jj z a

z bPnzjj

Zpkz DnX

jD0hZpk pjipjz D

bkjpjz D

Dn1XjD0

given n this gives

0BBBBBBBBB

p0zp1zp2z

1CCCCCCCCCA

0BBBBBBBBB

b00 b01 0 0 0 0

b10 b11 b12 0 0 0

b20 b21 b22 b23 0 0

bn2n1 0

1CCCCCCCCCA

0BBBBBBBBBB

p0zp1zp2zp3z

pn1zpnz

1CCCCCCCCCCA

0BBBBB

b00 b01 0 0 0

b10 b11 b12 0 0

b20 b21 b22 b23 0

1CCCCCA

0BBBBBBBB

p0zp1zp2zp3z

1CCCCCCCCA

0BBBBBBBB

1CCCCCCCCA

ŒMM D 1 ˝ 1 D

0BBBBB

a c 0 0

b a c 0

0 b a c0 0 b a

1CCCCCA

jcj2 jbj2 0 0 0 0 0

ŒZZ D 1 ˝ 1

0BBBBB

Z0 A0 0 0

0 Z1 A1 0

0 0 Z2 A2

0 0 0 Z3

1CCCCCA

ŒZkZk C AkA

k Ak1Ak1 D 0 k 1

AkZkC1 D Z

ŒZ0Z0 C A0A

0 D 1 ˝ 1

ker An D 0

ZkC1 D AkZkA1k k 0

kA z1k2 Dd1XkD0

jQkzj2jPzj2 (412)

deg Qk D k 0 k d 1

jPzj2 Dd1XkD1

jQkzj2

Zpzdnz D tr pAn

n p 2 CŒz

PnA HnA

Then detz An D Pnz

QnAn Akn k lt n

An WAn WA

pAA the trace

tr pAn tr pBn

Akn Bk

jD1Aj1

n An BnBkjn

jtrAkn Bk

limn1Œ

tr pAn tr pBn

kT T 1k lt 1

Consequently

kT T 1PnTk lt kPnTk

ŒTT D ˝

zkC1w`C1

hTkT`i D hTkN T

`N i k N 1 ` N or k N ` N 1

ENzw D 1 hTN w1 T

ENzw D QNzw

PNzPNw

Ezw D 1 1X

zkC1w`C1

ck`1k`D0 0 rankck`

1k`D0 minN n

N1k`D0 gt 0

ck` D hT`N T

1XkgtN or `gtN

jEzw ENzwj 2R2N

1XkgtN or `gtN

R0kC`C2

Zn z11 1

zD npn

znC1 C O1

Zn z11 Dn1XkD0

0BBBBB

a0 b1 0 0

c1 a1 b2 00 c2 a2 b3 0 0 c3 a3

1CCCCCA

respectively

0BBBBB

a0 c1 0 0

b1 a1 c2 00 b2 a2 c3 0 0 b3 a3

1CCCCCA

ŒTT D

0BBBBB

r 0 0 0 0 0 0 0

0 0 0 0

1CCCCCA

linear equations

This implies

Consequently

a1 D a2 D a3 D D a

hA z1 A w1iC

V W H1 ˚ H2 H ˚ H1 ˝ H2

with the property

A1 z11A2 z12

A z1A1 z11 ˝ A2 z12

A1 z1 ˝ I I ˝ A2 z1 D

A1 z1 ˝ II ˝ A2 z A1 z˝ II ˝ A2 z1 D

A1 z1 ˝ A2 z1ŒI ˝ A2 A1 ˝ I

A1 z11 ˝ A2 z12 D

ŒI ˝ A2 A1 ˝ I1A1 z1 ˝ I I ˝ A2 z11 ˝ 2

pA11pA22

pApA1˝IpI˝A2

I˝A2A1˝I 1 ˝ 2

WpA11 D

pAI ˝ A2 A1 ˝ I1pA11 ˝ 2

pA2 D 0

Dx D I ˝ A2 A1 ˝ I1x ˝ 2

A1 ˝ ID D DA1 (51)

kxk2 kDxk2 D kyk2

we obtain

Area˝2p dist˝1˝2

f dA DZ

Kf d f 2 O˝

h f gi D 1

ZHzwf zf wdzdw

Hf w D 1

ZHzwf zdz

pnz D nzn C Ozn1

qnz D nzn C Ozn1

1n D inf

deg f n1 kzn f k2

HnZn D nAnC1Hn D A

n Hn C nAnC1 nHn

HnZn H1

n An D nAnC1 nHnH1

dist conv K r

H1n ZnHnun D nun

n ZnHn Ank

n D H1n ZnHn An The

nC1 n D qn qn

whence

n D nAqnH1n nHqn

qnzC Ozn1

and consequently

we infer

nAqn D n1n

HnH1n f qn D f H1

n nHqn

Hh D s C t0 nt0 D 0

h D H1n nHqn

h D qn n

npnk2 D 0

npn for all n 0

limn1 kqn nn

Pnz D pnz

Qnz D qnz

which yield

hzn zki DX

jminnk

rnjrkj

respectively

zn zk DX

jminnk

snjskj

1j`D0 S D sj`

B D RR N D SS

pn DXkn

is therefore

C D R1S

ınm D Hpn pm DXk`

cnkHqk q`cm`

I D CHC

kqn pn

cnnk22 D

n1XkD0

H D I C LDI C L

TVn VnC1 TVn VnC1

w limnn D d1

dıa1 C d2

dıa2 C C dm

0BBBBB

T0 0 0 0

T1 0 0

0 T2 0 0 T3

1CCCCCA

dim Vn D n C 1d

tr pRn DnX

Zp dn D tr pRn

dim VnD d1

dpa1C d2

dpa2C C dm

61 General

78 6 Mother Bodies

Uz D 1

Zlog jz j d

Cz D 1

˝e D ˝ j˝j D 0 (61)

61 General 79

U D U in ˝e

U U in all C

jsuppj D 0

13energy DZ

Ud13 DZ

80 6 Mother Bodies

Rd ltR

Bal D ˝ (62)

dA 2 SL1˝ (63)

if D 12131 C 132 then D 131 D 132

82 6 Mother Bodies

f dA f 2 O˝

tC L D 0

f dA DZ˝

f d f 2 O˝ (64)

f dA DmX

nk1XjD0

ckjf jak (65)

ck0ıak

84 6 Mother Bodies

f dA D af 0C i f 1C f C1 f 2 O˝ (66)

i f 1C f C1 D iZ

xdx D i

u dA D au0CZ

h f gi D 1

Hzwf zgw dzdw (67)

1 Fzw D hkz kwi D 1

Hu vdu

h f gi D 1

86 6 Mother Bodies

f gL2˝ D 1

z 7 hkz 1i

zw 7 hkz kwi

z 7 kz

supp K (610)

C˝ D C on C n˝ (611)

equivalently

˚z D kz (612)

88 6 Mother Bodies

h dA DZ

Kh dA (613)

h DZ˝

Hzwkzw dAw

hkz i D hkz i

u D U U (614)

u D in ˝ (615)

u D jruj D 0 on ˝ (616)

u D (617)

Sz D Nz 4u

z (618)

uz D 1

4jzj2 jz0j2 2Re

Sd (619)

90 6 Mother Bodies

z˚z Sz D z˚z Nz u

z˚ Sd (620)

and AnAC

(C D ACnA

C R2 r2ıC1 D AnAC

C R2 r2ı1

andAnAC

uRz D 1

4jzj2 R2 R2 log

jzj2R2 (622)

(uCz D minACnA

(C D 1 uC D 1 u

92 6 Mother Bodies

Chapter 7Examples

71 The Unit Disk

h f gi D f 0g0

enk D 1

k C 1znNzk

One computes that

0 otherwise

94 7 Examples

72 The Annulus

For the annulus

Gzw D(

enz D8lt

R2r2 n lt 0

R2r2 n 0

f DXn2Z

cnen jj f jj2 DXn2Z

jcnj2 lt 1 (71)

f z DXnlt0

R2 r2zn C

R2 r2zn (72)

enzenw zw 2 ˝

2 P n Dg

jzj jwj gtgt 1

96 7 Examples

1 zS01 NwS0Q 1z 1Nw

D Cpzpw

qz Nw (73)

qz Nw D zd NwdQ1

Nw (74)

pz D zdQ 1z 0

1 zS0D zd1 Q 1z 0

b D 2m C d 2 (75)

S˝z D 1

731 The Ellipse

˝ D fz 2 C W x2

a2C y2

b2lt 1g

dx D 2ab

d13 D dx

Sz D a2 C b2

c2z 2ab

pz2 c2

Hzw D C

98 7 Examples

13n 13

as n 1

Sek D ekC1 k 0

essT D fr C 1

jj D 1g

pr (Fig 71)

0BBBBBBBB

0 r 0 0 0

1 0 r 0 0

0 1 0 r 0

0 0 0 0 r0 0 1 0

1CCCCCCCCA

r Recall that

Uncos D sinn C 1

d D dx

p4r x2

2pr lt x lt 2p

732 The Hypocycloid

zt D aeit C beid1t

100 7 Examples

a d 1b (77)

D a C b1d

a C bd

D W Qz Nz D 0

pz D zdQ1

z 0 D ˇ

H˝zw D C

q 1= N D 0 2 P (79)

H˝zw D C

102 7 Examples

74 Lemniscates 103

74 Lemniscates

zm 1Nzm 1 D R2m

Sz D m

szm 1C R2m

zm 1 (710)

dx D sin m=

xm 1C R2m

dx D sin=m

xm 1C R2m

plus rotations

104 7 Examples

g˝ez1 D 1

mlog jzm 1j log R

Rmjdzj on ˝

75 Polygons 105

2g˝e1 D 1

75 Polygons

uz D 1

2dist z˝e2

106 7 Examples

75 Polygons 107

108 7 Examples

u D minu1 u2

(u1z D 1

2Re z2

where 0 lt ˛ lt 2

u1z D 12Im ei˛z2

u3z D 12Im ei˛z2

u1z D 14x2 C y2 2xy

u1z D 14x2 y2 2xy

u3z D 14x2 y2 C 2xy

110 7 Examples

See Figs 78 and 79

Ca W jz aj2 D 1C a2

Cb W jz bj2 D 1C b2

uaz D 1

112 7 Examples

78 External Disk

114 7 Examples

79 Abelian Domains

h dA D cZ a

ah dx C

ckhak (712)

f D A log1C ˛

1 ˛C B (713)

h dA D AZ a

ah dx C 2˛AB h0

116 7 Examples

f D A log1C ˛

1 ˛ C B

1C ˇ22 (714)

h D H0 whereby

h dA D 1

H0zdzdNz D 1

H f df 1= N

uz D 1

2jzj2 R2 log

jzj2R2

81 Bergman Space

with inner product

h f giK ˝ D 1

Kzwf zgwdAzdAw

Kzwf zgwdAzdAw D 1

f w D 1

Pf w D 1

f zKzw dAz f 2 L2˝

PNf z D 1

Kzwf w dAw

fn w D wn C Rnw1

T h D P h h 2 H2T

Phz D 1

T D ˝ essT D

Q D a1S C a2S2 C

ŒT T D a21ŒS

SC ŒQQ a211 ˝ 1

r1T r D a1S C a0

indT D 1 2 ˝

a0 a1 a2 a1 a0 a10 a1 a0

T n D nT n

Tfn 1 D wn n 0

FnT 1 D wn n 0

p eitq eis

1 eistdtds

0BBBBB

c0 2c1 3c2 4c3 1 c0 c1 c20 1 c0 c1 0 0 1 c0

1CCCCCA

C C an

0 D lim supn1

jcnj1=n

W fz W jzj gt 0g C

ŒTT D TT TT 0

hTTx xi hTTx xi

or equivalently

kTxk kTxk x 2 H

ŒYx D axx bx

ŒXYx D bx

ZIbyydy

ŒTT D 2iŒXY 0

kTnk D kTkn n 1

kTTnxk kTnC1xk

whence

kTTnk kTnC1k

Consequently

kTknC1 kTnC1k

ŒI T1 I T1 D

kI T1k D 1

dist T

WT D convT

hTx xi D

kŒTTk Area T

TraceŒTT mT

Area T

ŒTT D ˝

T zT wT z1T w1

detT zT wT z1T w1 D

1 hT z1T w1 i D

1 hT w1 T z1i

detT wT zT w1T z1 D

1C hT w1T z1 i D

1C hT z1 T w1i

z w jzj jwj gt kTk

h f T gTi D 1

f ugvdudv

Eu v (A2)

hgT f Ti D 1

f ugvEu vdudv (A3)

f wd Areaw

w zj2 kf k1kf k1

h T z1i D 1

f wd Areaw

kT z1k 1 z 2 C

kk2 D 1

f wd Areaw

traceŒ pTT qTT D 1

T f D Pf f 2 H2

z D Czz z 2 T

T D TTC

TCn D PnTC

Pn D PnTC T

n D PnTPn D T

Note that TCn T

n D PnTCT

G D exp1

logzdz

RTCz dz

iz respectively 12

RTz dz

GnC1 D detTCn det T

Tn D PnTPn D PnTTC

Pn D PnTCT1

TCn PnT1

Therefore

det Tn

GnC1 D det Tn

det TCn det T

D PnT1C

det T1C

D det TTC

D detTT1

det Tn

JlogC logdA

Historical Notes

Glossary

P D C [ f1g

˝c D C n˝

j˝j D Area˝

f g2˝ D f gL2˝ D 1

140 Glossary

f gduality D 12 i

Hf w D 1

Hzwf zdAz w 2 ˝

See (37)

Gf w D 1

Gzwf zdAz w 2 ˝e

See (333)

References

142 References

References 143

144 References

Sci (4) 20(3) 323ndash339 (1993)

References 145

148 Index

Editorial Policy

Preface

Contents

1 Introduction

22 Moments
6 Mother Bodies
61 General
7 Examples
71 The Unit Disk
72 The Annulus
731 The Ellipse
732 The Hypocycloid
74 Lemniscates
75 Polygons
78 External Disk
79 Abelian Domains
81 Bergman Space
Historical Notes
Glossary
References
Index

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