Compact Hankel Forms on Planar Domains

Compl. anal. oper. theory 3 (2009), 471–499c© 2008 Birkhauser Verlag Basel/Switzerland1661-8254/020471-29, published online October 24, 2008

DOI 10.1007/s11785-008-0094-6

Complex Analysisand Operator Theory

Compact Hankel Forms on Planar Domains

Vasiliy A. Prokhorov and Mihai Putinar

To Bjorn Gustafsson on the occasion of his sixtieth birthday

Abstract. A Hankel form on a Hilbert function space is a bounded, symmet-ric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f .We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankelforms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected pla-nar domains. In the case of the unit disk, we investigate the asymptotic ofsome measures constructed by eigenfunctions of Hankel operators with certainMarkov functions as symbols.

Mathematics Subject Classification (2000). Primary 41A20; Secondary 30E10,47B35.

Keywords. Bilinear symmetric form, singular number, meromorphic approxi-mation, rational approximation.

1. Introduction

The present article offers a unifying perspective on some meromorphic approxi-mation theory questions referring to a planar open set G (with some boundaryregularity) and a (positive) Borel measure μ, compactly supported by G. By spe-cializing an abstract study of the structure of a Hilbert space endowed with asymmetric bilinear form, we derive estimates and other qualitative features of:the singular numbers associated to Hankel operators with Markov functions assymbols, doubly orthogonal systems of analytic functions and Hadamard type de-terminants. The cases of Hardy and Bergman spaces are treated in parallel andwith the same tools.

To begin with, we trace some history and main facts related to the classicalapproximation theory questions we are concerned with. The Markov function (also

The second author was partially supported by a grant from the National Science Foundation.

472 V. A. Prokhorov and M. Putinar Comp.an.op.th.

known as the Cauchy transform) of a positive Borel measure μ:

μ(z) =1π

∫dμ(x)z − x

(1)

is an analytic function defined outside the closed support suppμ. Quite a few recentinvestigations in Approximation Theory deal with estimates of natural approxi-mations of μ restricted to ∂G. We mention the papers of Baratchart, Saff andthe first author [6] and [8] (see also [7] and [9]), where the convergence of thebest meromorphic approximants of the Markov function in the space Lp(Γ), thestrong asymptotic of errors in best meromorphic approximation, and the limitingdistribution of poles of the best approximants are obtained in the case when G isthe open unit disk, 1 ≤ p ≤ ∞, and the positive Borel measure μ with supportsuppμ = [a, b] ⊂ (−1, 1) satisfies Szego’s condition on [a, b]. We also single outthe papers of Braess [14] and J.-E. Andersson [4] concerning the best meromor-phic approximation of Markov functions. Convergence and asymptotic behavior oferrors in the best rational approximation of Markov functions in the space L2(Γ)were investigated by Baratchart, Stahl and Wielonsky [10–12]. Problems relatedto the best uniform rational approximation of Markov functions were consideredby Gonchar [24] and W. Barrett [13]. One of the oldest and classical results ofthe rational approximation of analytic functions is Markov’s theorem (see, forexample, [34] and [5]). This theorem states the uniform convergence of Pade ap-proximants (locally the best rational approximants to a given power series) of theMarkov function. For more results about convergence of Pade approximants ofMarkov functions see the works of Gonchar [23], Rakhmanov [51], Gonchar andSuetin [27], Gonchar and Lopez [25], Totik [59], Stahl and Totik [56].

There are good reasons to consider the case when the Markov function is ap-proximated by meromorphic functions in the space L∞(Γ), where Γ is the boundaryof the domain G. Methods of meromorphic approximation can be applied in thiscase to obtain results characterizing the degree of rational approximation of suchanalytic functions. In this connection we mention the papers of Parfenov [36] andthe first author [42, 43]. Here is the point where the (well developed) theory ofHankel operators and their truncations (the so called Adamyan–Arov–Krein the-ory) comes naturally as a theoretical support. For any continuous function f on Γthe Hankel operator Af = Af,G : E2(G) → E⊥

2 (G) is defined by

Af (ϕ) = P−(ϕf) , ϕ ∈ E2(G) ,

where E2(G) is the Smirnov class of analytic functions on G with boundary valuesidentified as a subspace of L2(Γ), P− is the orthogonal projection of L2(Γ) ontoE⊥

2 (G). In the case G is a disk, the well-known theorem of Adamyan–Arov–Kreın(see [1, 2]) establishes a connection between the singular values of the Hankeloperator and a complex approximation quantity. Specifically, denote by

Δn,∞(f ;G) = infh∈Mn,∞(G)

||f − h||∞

Vol. 3 (2009) Compact Hankel Forms on Planar Domains 473

the error in the best meromorphic approximation of f in the space L∞(Γ) bymeromorphic functions from the class

Mn,∞(G) ={h = p/q : p ∈ E∞(G), q is a polynomial, deg q ≤ n, q �≡ 0

}.

According to the Adamyan–Arov–Kreın theorem, the n-th Hankel singular valuesn(Af ) = sn(Af ;G) for Af (i.e., the eigenvalues of |Af | = (A∗

fAf )1/2) is identifiedwith the error Δn,∞(f ;G). We also state an extension of this result (see [41]) to thecase when G is a bounded domain whose boundary Γ consists of N closed analyticJordan curves and where the equality is replaced by upper and lower bounds:

Δn+N−1(f ;G) ≤ sn(Af ) ≤ Δn(f ;G) for n ≥ N − 1 . (2)

In this case the following analog the Weyl–Horn inequality [22] was proved in [44]:∣∣∣∣det(∫

Γ

(ϕiϕj)(ξ)f(ξ)dξ

)∣∣∣∣ ≤n∏

k=0

sk(Af ) det(〈ϕi, ϕj〉) . (3)

Above the functions ϕ0, . . . , ϕn belong to the Smirnov class E2(G) and 〈ϕi, ϕj〉 isthe inner product in the space L2(Γ). One of the direct corollaries of this result isthe estimate (see [45]):

n∏k=0

sk(Af ;G)sk(Af ;G1)

≤n∏

k=0

s2k(J) , (4)

where sk(Af ;G) and sk(Af ;G1) are the singular numbers of the Hankel operatorsAf,G and Af,G1 , G1 ⊂ G, f is an analytic function with singular points in G1 andJ : E2(G) → L2(ds, ∂G1) is the restriction operator. By means of this inequalitythe first author has obtained results concerning the best rational approximationρn(f ;E) to an analytic function f in the uniform metric on a compact set E, byrational functions of degree at most n. In [45] the following result characterizingthe relationship between ρn(f ;K) and ρn(f ;E) in the case when complements ofthe compact sets K and E are connected, K is a subset of the interior Ω of E,and f is analytic in Ω and continuous on E is proved (see also [32] for a moregeneral situation):

lim supn→∞

(n∏

k=0

ρk(f ;K)ρk(f ;E)

)1/n2

≤ exp(− 1/C(∂E,K)

), (5)

where C(∂E,K) is the capacity of the condenser (∂E,K). The inequality (5)implies Parfenov’s estimates of the degree of rational approximation [36] (referringto case when the compact set E where the given analytic function is approximatedis a continuum with connected complement) and the estimates obtained by thefirst author [42] (in the case of an arbitrary compact set E):If f is analytic on C \ F, and E ∩ F = ∅ then

lim supn→∞

(n∏

k=0

ρk(f ;E)

)1/n2

≤ exp(− 1/C(E,F )

). (6)


The contents of the present article is the following. In Section 2 we presentabstract versions of Weyl–Horn and Ky Fan inequalities (see, for example, [22]) forcompact bilinear symmetric forms defined on a complex separable Hilbert space H.Section 3 is devoted to Hankel operators on multiply connected domains withsmooth, real analytic boundary and some relevant properties of the compact bi-linear Hankel form

[u, v] =∫

Γ

(uvf)(ξ)dξ =∫

E

(uv)(x)dμ(x) , u, v ∈ E2(G) .

In particular, we investigate the connection between the Hankel operator con-structed from the Markov function μ and the restriction operator J : E2(G) →L2(μ,E), where E = suppμ. In Section 4 we return to the classical setting of adisk G and describe the asymptotic behavior of some measures associated withthe Hankel operator Af : E2(G) → E⊥

2 (G) whose symbol is a Markov function μ,where suppμ = [a, b] ⊂ (−1, 1) and μ′ = dμ/dx > 0 a.e. on [a, b].

Section 5 marks the transition from Hardy space to the Bergman space associ-ated to G. We consider a compactly supported measure μ which is gravi-equivalentto G (i.e. μ and the uniformly distributed mass on G have the same logarithmic po-tentials outside G). The spectral theory of the Friedrichs operator of the domain G(an integral operator considered by Friedrichs in questions of linear elasticity) willturn out to be intimately related to some qualitative features of the measure μ.For a recent introduction, using modern language, to the analysis of Friedrichs’operator see [49,50]. In this article we reveal a remarkable parallelism between theintegral operators (Hankel, respectively Friedrichs) appearing in the two scenarios.

2. Abstract framework

Let H be a complex separable Hilbert space and let [x, y] be a compact bilinearsymmetric form defined on H. In other terms, there exists a complex antilinearcompact operator T : H −→ H representing this form:

[x, y] = 〈x, Ty〉 = 〈y, Tx〉 , x, y ∈ H .

Let C : H −→ H be an arbitrary antilinear, isometric conjugation on H:

〈Cx, y〉 = 〈Cy, x〉 , C2 = I .

Then it is easy to see that there exists an orthonormal basis {en}n of H with theproperty:

C∑

n

cnen =∑

n

cnen .

In its turn, the conjugation C defines a symmetric bilinear form

{x, y} = 〈x,Cy〉 = 〈y, Cx〉 .

The operator S = CT is what we call today a C-symmetric operator, thatis S is symmetric with respect to the form {., .}:

{Sx, y} = 〈CTx,Cy〉 = 〈y, Tx〉 = 〈x, Ty〉 = {x, Sy} .


Another way to check the C-symmetry of S is the identity S∗ = CSC. For a recentaccount of C-symmetric operators we refer to [20].

Complex symmetric operators have appeared early, in the work of Takagi onbounded analytic interpolation, [57]. In our setting, his main observation is thatthe antilinear eigenvalue problem:

Tun = λnun , λn ≥ 0 ,

detects all (unit) eigenvectors un and eigenvalues λ2n of the self-adjoint operator

T 2. Moreover, in case√

T 2 is compact, the following strongly convergent {., .}-diagonalization holds:

Sϕ = CTϕ =∑

n

λn〈ϕ, un〉Cun , ϕ ∈ H ,

or equivalentlyTϕ =

∑n

λn〈un, ϕ〉un , ϕ ∈ H .

Note that the operator T is antilinear, so the phase of λn in the eigenvalue problemis at will. And we choose it for convenience to be zero. Also, remark that theoperators S = CT and T are not self-adjoint, so the above decomposition is anon-trivial generalization of the spectral theorem.

When speaking about the bilinear form [., .] only, it will be natural to call thenumbers λn the characteristic values of [., .] (with respect to the fixed Hilbert spacestructure of H). These numbers, and the associated eigenvectors, are characterizedby the double orthogonality conditions:

[un, um] = λnδmn , 〈un, um〉 = δmn , (7)

where δmn is Kronecker’s symbol. We will arrange the eigenvalues in nonincreasingorder

λ0 ≥ λ1 ≥ · · · ≥ 0 .

The following variant of Weyl–Horn estimate will be important for our study.

Theorem 1. Let [., .] be a compact bilinear symmetric form on a complex Hilbertspace H and let {λk}k denote its characteristic values. Let g0, . . . , gn be a systemof vectors in H. Then for any nonnegative integer n,∣∣ det([gi, gj ])

∣∣ ≤ λ0 . . . λn det(〈gi, gj〉) . (8)

Proof. Let {uk}k be the orthonormal system of eigenvectors of the associatedantilinear operator T . Write

gi =∞∑

k=0

cikuk , 0 ≤ i ≤ n .

Then[gi, gj ] =

∑k

cikcjk[uk, uk] =∑

k

λkcikcjk .


Therefore

det([gi, gj ]) =1

(n + 1)!

∑k0,...,kn

λk0 . . . λkn

(det(cikj

))2

≤ λ0 . . . λn1

(n + 1)!

∑k0,...,kn

∣∣ det(cikj)∣∣2

= λ0 . . . λn det

(∑k

cikcjk

)

= λ0 . . . λn det(〈gi, gj〉) . �

Next we state a compact bilinear symmetric form variant of Ky Fan inequality(see, for example, [22]).

Theorem 2. Let [ · , · ] be a compact bilinear symmetric form on a complex Hilbertspace H and let {λk}k denote its characteristic values. Then for any orthonormalsystem g0, . . . , gn of vectors in H∣∣∣∣∣

n∑i=0

[gi, gi]

∣∣∣∣∣ ≤ λ0 + · · · + λn .

Proof. As above, let {uk}k be the orthonormal system of eigenvectors of the asso-ciated antilinear operator T , corresponding to the system of characteristic values{λk}k. We have

gj =∞∑

k=0

cikuk , 0 ≤ j ≤ n ,

〈gi, gj〉 =∑

k

cikcjk = δij , (9)

and[gi, gj ] =

∑k

λkcikcjk .

It is easy to see that∣∣∣∣∣n∑

i=0

[gi, gi]

∣∣∣∣∣ =

∣∣∣∣∣n∑

i=0

∑k

λkc2ik

∣∣∣∣∣ ≤n∑

i=0

∑k

λk|cik|2 .

Let us consider the following polynomial of degree n + 1:

P (λ) = det

(∑k

(λk − λ)cikcjk

). (10)

As above, one finds

P (λ) =1

(n + 1)!

∑k0,...,kn

(λk0 − λ) . . . (λkn− λ)|det(cikj

)|2 .


From this, by (9) and (10), we infern∑

i=0

∑k

λk|cik|2 =1

(n + 1)!

∑k0,...,kn

(λk0 + · · · + λkn)|det(cikj

)|2

≤ (λ0 + · · · + λn)1

(n + 1)!

∑k0,...,kn

|det(cikj)|2

= (λ0 + · · · + λn) det(〈gi, gj〉) = λ0 + · · · + λn . �

Let X and Y be the Hilbert spaces, and let A : X → Y be a compactoperator. We denote by {sn(A)}, n = 0, 1, 2, . . . , the sequence of singular numbersof the operator A (i.e. the sequence of eigenvalues of the operator |A| = (A∗A)1/2,where A∗ : Y → X is the adjoint of A). Throughout this work we will adopt thenon-increasing ordering: s0(A) ≥ s1(A) ≥ · · · ≥ sn(A) ≥ · · · .

For a given antilinear isometric conjugation C, a C-symmetric operator Tadmits the polar decomposition

T = CJ |T | ,where J is another antilinear isometric conjugation, commuting with |T |. Andconversely, every such product is C-symmetric (see for a proof [20] part II). If Tis in addition compact, then a simple variant of Adamyan–Arov–Krein Theoremholds. Namely, for every positive integer N , the following (N+1)-rank perturbationof T is optimal:

‖T − TN‖ ≤ sN+1(T ) ,

where TN is C-symmetric of the form:

TN = CJπN+1|T |πN+1 ,

and πN+1 denotes the orthogonal spectral projection of |T | corresponding to theinterval [0, sN+1(T )], see for details [20].

3. Hankel operators on multiply connected domains

This section deals with the interplay between Hankel operators on multiply con-nected domains and the best meromorphic approximation of their symbols.

3.1. The modulus a Hankel operator attached to a Markov function

Let G denote a bounded domain in the complex plane C. We assume that bound-ary Γ of G consists of a finite number closed analytic Jordan curves and is positivelyoriented with respect to G. Here, and throughout this paper, Lp(Γ), 1 ≤ p < ∞,denotes the Lebesque space of functions ϕ measurable on Γ, endowed with thenorm

||ϕ||p =(∫

Γ

|ϕ(ξ)|p|dξ|)1/p

< ∞ .


Let

〈ϕ,ψ〉 =∫

Γ

ϕ(ξ)ψ(ξ)|dξ| , ϕ, ψ ∈ L2(Γ) ,

be the inner product in the Hilbert space L2(Γ). Denote by L∞(Γ) the space ofessentially bounded on Γ functions ϕ with the norm

||ϕ||∞ = ess supΓ

|ϕ(ξ)| < ∞ .

Let Ep(G), 1 ≤ p ≤ ∞, be the Smirnov class of analytic functions on G. Weconsider Ep(G) as a closed subspace of the space Lp(Γ). The condition∫

Γ

ϕ(ξ)dξ

ξ − z= 0 for all z ∈ C \ G ,

is necessary and sufficient for a function ϕ ∈ L1(Γ) to be the boundary valueof a function in the Smirnov class E1(G) (see [40] and [58] for more details aboutthe classes Ep(G)). In this section we consider the situation when G is a domainwith a boundary consisting of a finite number closed analytic Jordan curves. Itis the well-known fact (see, for example, [58]) that in this case the Smirnov classEp(G), 1 ≤ p ≤ ∞, coincides with the Hardy space Hp(G) of analytic functions onG (see [21,31,61] for more details about Hardy spaces).

Let us represent the space L2(Γ) as the direct sum

L2(Γ) = E2(G) ⊕ E⊥2 (G) ,

where E⊥2 (G) is the orthogonal complement of E2(G) in L2(Γ). Recall an auxiliary

result giving necessary and sufficient conditions for a function a ∈ L2(Γ) to be anelement of the subspace E⊥

2 (G) (see [41]) :Let a ∈ L2(Γ), then a ∈ E⊥

2 (G) if and only if there exists a function b ∈E2(G) such that

a(ξ)|dξ| = b(ξ)dξ a.e. on Γ . (11)Remark that the boundary matching condition in the above statement is also

equivalent toa(ξ)dξ = b(ξ)|dξ| a.e. on Γ . (12)

Let f be continuous on Γ. The Hankel operator Af : E2(G) → E⊥2 (G) is defined

byAf (ϕ) = P−(ϕf) , ϕ ∈ E2(G) ,

where P− is the orthogonal projection of L2(Γ) onto E⊥2 (G). It is not hard to see

that Af is a compact operator.Let μ be a positive Borel measure with (closed) support suppμ = E ⊂ G.

Henceforth all measure supports are assumed by convention to be closed. Denoteby L2(μ,E) the Hilbert space with respect to the inner product

〈ϕ,ψ〉2,μ =∫

E

ϕ(x)ψ(x)dμ(x) , ϕ, ψ ∈ L2(μ,E) ,

and the norm ||ϕ||2,μ.


One of our goals is to investigate the connection between the Hankel operatorAf with symbol f given by the Markov function (the Cauchy transform) of themeasure μ:

f(z) =1

2πi

∫E

dμ(x)z − x

=12i

μ , (13)

and the restriction operator J : E2(G) → L2(μ,E) defined by:

J(ϕ) = ϕ|E , ϕ ∈ E2(G) .

In view of Montel’s Theorem, the operator J is compact. According to some au-thors, J is also known as the embedding operator (of E2(G) into L2(μ,E)).

Define the bilinear symmetric Hankel form

[u, v] =∫

Γ

(uvf)(ξ)dξ , u, v ∈ E2(G) .

If the function f is the Cauchy transform of a measure μ, compactly supportedby G, then we derive, via Cauchy’s theorem, a second representation for the sameform:

[u, v] =∫

E

(uv)(x)dμ(x) . (14)

Thus the form [., .] is compact and the abstract framework outlined in thepreliminaries applies.

To better understand the representing operators, we start by the very defini-tion of the Hankel operator Afu = uf − w ∈ E⊥

2 (G), w ∈ E2(G). Hence, by (12),there exists an element Bf (u) ∈ E2(G) characterized by the boundary identity:

(Afu)(ξ)dξ = (uf − w)(ξ)dξ = Bf (u)(ξ)|dξ| a.e. on Γ . (15)

It is clear that the dependence of Bf on both f and u is antilinear. Directcomputation yields, for a pair of functions u, v ∈ E2(G):

[u, v] =∫

Γ

(uvf)(ξ)dξ =∫

Γ

v(ξ)Bf (u)(ξ)|dξ| = 〈v,Bfu〉 = 〈u,Bfv〉 . (16)

Thus, the antilinear operator representing the Hankel form [., .] with respectto the Hardy–Smirnov sesquilinear form is precisely Bf .

Theorem 3. For the Markov function f = 12i μ, where μ is a positive Borel measure

with support suppμ = E ⊂ G, we have

|Af |2 = A∗fAf = (J∗CJ)2 , (17)

where C(ϕ) = ϕ, ϕ ∈ L2(μ,E). Moreover, for n = 0, 1, 2, . . . ,

n∏k=0

sk(Af ) ≤n∏

k=0

s2k(J) . (18)


Proof. For the Markov function f,

Bf = J∗CJ , (19)

where J : E2(G) −→ L2(μ,E) is the restriction operator and Cϕ = ϕ,ϕ ∈L2(μ,E), is the complex conjugation. Indeed, for a pair of functions u, v ∈ E2(G),

〈v,Bfu〉 = [u, v] =∫

E

(uv)(x)dμ(x) = 〈Jv,CJu〉2,μ = 〈v, J∗CJu〉 .

So, we have (see (15))

〈Afu,Afv〉 =∫

Γ

(Afu)(ξ)Af (v)(ξ)|dξ| = 〈Bfv,Bfu〉2,μ =⟨u,B2

fv⟩2,μ

.

That isA∗

fAf = B2f = (J∗CJ)2 .

In order to prove inequality (18) between the singular numbers of Af andthose of J we appeal to Theorem 1. Namely, let sk(Af ) be the singular numbers

of Af , that is of√

B2f , and let Qk be the associated orthonormal eigenvectors:

BfQk = sk(Af )Qk (20)

and〈Qi, Qj〉 = δij , [Qi, Qj ] = δijsj(Af ) . (21)

By Theorem 1 applied to the conjugation C we find:n∏

k=0

sk(Af ) = det([Qi, Qj ]) = det(〈JQi, CJQj〉2,μ

)

≤ det(〈JQi, JQj〉2,μ

)≤

n∏k=0

sk(J)2 .

This completes the proof of the theorem. �3.2. Integral formulas

Next we establish simple integral formulas relating the Hankel operator Af andthe embedding operator J . Denote by K(z, ξ) the reproducing kernel of the spaceE2(G). That is, for all ξ ∈ G, K( · , ξ) ∈ E2(G), K(z, ξ) = K(ξ, z), and for anyϕ ∈ E2(G)

ϕ(z) =∫

Γ

ϕ(ξ)K(z, ξ)|dξ| . (22)

For instance, if G is the unit disk {z : |z| < 1}, then

K(z, ξ) =1

2π(1 − zξ). (23)

It is easy to see that the adjoint J∗ : L2(μ,E) → E2(G) of the restriction opera-tor J admits the integral kernel:

J∗(w)(z) =∫

E

w(t)K(t, z)dμ(t) , w ∈ L2(μ,E) . (24)


Therefore, for ϕ ∈ E2(G), we obtain

Bf (ϕ)(z) = (J∗CJ)(ϕ)(z) =∫

E

ϕ(t)K(z, t)dμ(t) (25)

and consequently

(A∗fAf )(ϕ)(z) = (J∗CJ)2(ϕ)(z) =

∫E

∫E

ϕ(t)K(z, ξ)K(t, ξ)dμ(t)dμ(ξ) .

Moreover, we mention the following integral equation satisfied by eigenfunc-tions Qn (see (20), (25)):

sn(Af )Qn(z) =∫

E

Qn(t)K(z, t)dμ(t) .

3.3. Symmetric domains

Let us consider the case when G is a domain symmetric with respect to the realaxis and μ is a positive Borel measure with support E ⊂ G ∩R. It is not hard toprove that in this case

A∗fAf = (J∗J)2 = (J∗CJ)2 (26)

andsn(Af ) = s2

n(J) , n = 0, 1, 2, . . .

(see also [6, 7]). Indeed, let U : E2(G) → E2(G) be the antilinear isometric conju-gation on E2(G) defined by the formula

(Uϕ)(z) = ϕ(z) , ϕ ∈ E2(G) , z ∈ G .

It follows directly from the definition of U that

J = CJU

andJ∗J = J∗CJU . (27)

For any pair of functions u, v ∈ E2(G),

〈J∗Ju, v〉 = 〈J∗CJUu, v〉 = 〈CJUu, Jv〉2,μ

= 〈CJv, JUu〉2,μ = 〈J∗CJv, Uu〉= 〈v, UJ∗CJu〉 .

That isJ∗J = UJ∗CJ .

Since U2 = I, from this and (27) we obtain (26).

4. Limit theorems in the disk

We specialize in this section to the case of a disk and a measure μ compactlysupported by a diameter. This is the oldest and best understood framework forthe related rational approximation problems we are dealing with.


4.1. Preliminaries

Let G = {z : |z| < 1} be the unit disk and μ be a positive Borel measure withsupport suppμ = E ⊂ G∩R. We assume that the support of μ contains infinitelymany points.

First we recall some known facts and introduce the necessary terminology.Denote by A2 the restriction on E of the closed unit ball of the Hardy–Smirnovclass E2(G):

A2 ={ϕ ∈ E2(G) : ||ϕ||2 ≤ 1

}.

For any positive integer n let Bn be the class of all Blaschke products of degree n.It is proved by S. D. Fisher and C. A. Micchelli (see [17] and [18]) that

dn(A2, L2(μ,E)

)= dn

(A2, L2(μ,E)

)= δn

(A2, L2(μ,E)

)= inf

B∈Bn

supϕ∈A2

||ϕB||2,μ = sn(J) , (28)

where dn, dn and δn are the Kolmogorov, Gelfand and linear n-widths of A2 inthe space L2(μ,E) (see, for example, [39]). S. D. Fisher and C. A. Micchelli alsoshowed [17, 18] that the eigenvalues of J∗J : E2(G) → E2(G) are simple, thecorresponding eigenspaces are one dimensional and that every eigenfunction Qn

has exactly n zeros in G. It is proved in [6] (see also (26)) that

A∗fAf = (J∗J)2

andsn(Af ) = s2

n(J) , n = 0, 1, 2, . . . ,

where f = 12i μ.

For a Bergman space analog and some alternate proofs of the above resultssee [29].

The eigenvalue equation

J∗JQn = s2n(J)Qn = sn(Af )Qn

implies〈Qi, Qj〉2,μ = s2

i (J)〈Qi, Qj〉 = s2i (J)δi,j .

Moreover, the same eigenfunction equation yields, when applied to the reproducingkernel of the disk:

s2n(J)Qn(z) =

12π

∫E

Qn(x)dμ(x)1 − xz

. (29)

This shows that every eigenfunction Qn analytically extends across the boundaryof the disk, to the Schwarz symmetric E−1 of the support E.

Due to the invariance of the pair (G,E) with respect to the symmetry on thereal line, one finds that Qn is a real valued function on E. Consequently, equation

sn(Af )|Qn(z)|2 =12π

∫E

1 − x2

|z − x|2 Q2n(x)dμ(x) , z ∈ Γ , (30)


follows. And similarly ∫E

Qi(x)Qj(x)dμ(x) = δijsj(Af ) (31)

holds. Furthermore, relation (30) implies

C1 ≤ |Qn(z)| ≤ C2 , z ∈ Γ , (32)

where

C1 = minx∈E,t∈Γ

1 − x2

2π|t − x|2and

C2 = maxx∈E,t∈Γ

1 − x2

2π|t − x|2 . (33)

Let x1,n, . . . , xn,n denote the zeros of Qn (all contained in the disk G). Thenthe following orthogonality relations are valid (see [6]):∫

E

xν

w∗n(x)

Qn(x)dμ(x) = 0 , ν = 0, . . . , n − 1 , (34)

where w∗n(z) =

∏nk=1(1 − xk,nz). Indeed, for any function ϕ ∈ E2(G) we have∫E

ϕ(x)Qn(x)dμ(x) = sn(Af )∫

Γ

ϕ(t)Qn(t) |dt| . (35)

Using ϕ(t) = tν/w∗n(t) in (35), we obtain (34). We also remark that from the

orthogonality relations (34) it follows that the zeros of Qn lie in the smallestclosed interval containing E. Henceforth we denote sn = sn(Af ) for all n.

4.2. Convergence of measures associated with Af

By keeping the notation introduced before, we describe in this subsection theasymptotic behavior of the measures

1(n + 1)!

(det

(Qi(xj)

))2 dμ(x0) . . . dμ(xn)s0 . . . sn

associated with the Hankel operator Af . We assume that the symbol f is theMarkov function

f(z) =1

2πi

∫dμ(x)z − x

=12i

μ(z) ,

suppμ = [a, b] ⊂ (−1, 1) and μ′ = dμ/dx > 0 a.e. on [a, b]. For a motivation whysuch limits are of interest (connection with distribution of eigenvalues of randommatrices, the k-point correlation functions), see [16,35].

First, we briefly recall some facts from potential theory. Denote by g(z, ξ) theGreen function of the open unit disk G with pole at ξ ∈ G:

g(z, ξ) = log∣∣∣∣1 − ξz

z − ξ

∣∣∣∣ .


For a positive Borel measure σ with support suppσ at G, we define the Greenpotential of σ

V σg (z) =

∫g(z, ξ)dσ(ξ)

and the energy Ig(σ) of σ with respect to the Green potential

Ig(σ) =∫

V σg (z)dσ(z) =

∫ ∫g(z, ξ)dσ(ξ)dσ(z) .

Let E be a compact subset of G. Consider the following extremal problem:

1C(E,Γ)

= infσ

Ig(σ) ,

where the infimum is taken in the class M(E) of all positive unit Borel measures σwith support suppσ ⊂ E. The extremal constant C(E,Γ) is called the capacityof the condenser (E,Γ). It is a well known fact (see, for example, [33]) that ifthe logarithmic capacity does not vanish, cap(E) > 0, then there exists a uniqueminimizing measure ω ∈ M(E) for which the infimum is attained:

1C(E,Γ)

= Ig(ω) .

The measure ω is called the Green equilibrium measure. In the case when E is acontinuum, it turns out that

ρ = e1/C(E,Γ)

is a conformal invariant, known as the modulus of the double-connected domainG \ E.

Example. In the case when E = [a, b] ⊂ (−1, 1) the following formula holds (seefor example [33]):

C(E,Γ) =2K ′

πK,

where

K =∫ 1

0

dx√(1 − x2)(1 − k2x2)

, k2 =(1 − a2)(1 − b2)

(1 − ab)2,

and K ′ is the corresponding elliptic integral for k′ =√

1 − k2. The Green equilib-rium measure is

dω(x) =k1dx√

(x − a)(b − x)(1 − ax)(1 − bx), x ∈ [a, b] ,

where

k1 =(1 − ab)

2K ′ .

We now return to the conditions imposed at the beginning of this section,namely that suppμ = E = [a, b] ⊂ (−1, 1) and μ′ = dμ/dx > 0 a.e. on [a, b]. In


this case it can be shown, using results of Gonchar and Rakhmanov [26], plus theextremal formulas (28) and also (32), that

limn→∞

s1/nn =

1ρ2

(36)

and1n

n∑k=1

δxk,n

∗→ dω as n → ∞ .

Above x1,n, . . . , xn,n are the zeros of Qn in G and δξ denotes the Dirac measurewith support at ξ. In view of (36) we obtain

limn→∞

(s0s1 . . . sn)1/n2=

1ρ

. (37)

Our next aim is to find an estimate from above of the determinant det(Qi(tj)).

Lemma 1. For any t0 ∈ G, . . . , tn ∈ G,

|det Qi(tj)| ≤ Cn+12 (n + 1)!

∏0≤i<j≤n

∣∣∣∣ ti − tj1 − tjti

∣∣∣∣ , (38)

where C2 is given by (33).

Proof. According to (32), we find∣∣ det(Qi(tj)

)∣∣ ≤ Cn+12 (n + 1)! (39)

for t0 ∈ Γ, . . . , tn ∈ Γ. Let us consider the function

dn(t0, . . . , tn) = log |det Qi(tj)| −∑

0≤i<j≤n

log∣∣∣∣ ti − tj1 − tjti

∣∣∣∣ ,

which is subharmonic function in G, with respect to each variable ti, when theremaining variables tj are fixed. By the maximum principle of subharmonic func-tions

dn(t0, . . . , tn) ≤ log(Cn+1

2 (n + 1)!)

whenever t0 ∈ G, . . . , tn ∈ G, From this we obtain (38). �

We focus now on the following theorem (compare to [16] and [30]).

Theorem 4. Let suppμ = [a, b] ⊂ (−1, 1) and μ′ = dμ/dx > 0 a.e. on [a, b]. Thenfor any continuous function u : E → R

limn→∞

(1

(n + 1)!

∫E

. . .

∫E

e∑ n

i=0 u(xi)(det Qi(xj)

)2 dμ(x0) . . . dμ(xn)s0 . . . sn

)1/n

= e∫

Eu(x)dω(x) .


Proof. Taking into account (31), we find

s0s1 . . . sn =1

(n + 1)!

∫E

. . .

∫E

(det Qi(xj)

)2dμ(x0) . . . dμ(xn) . (40)

Denote by dβn the positive Borel measure with support suppβn = E × · · · × Egiven by the formula:

dβn(x0, . . . , xn) =1

(n + 1)!

(det

(Qi(xj)

))2 dμ(x0) . . . dμ(xn)s0 . . . sn

.

It follows from (40) that ∫E×···×E

dβn = 1 .

Let

Vn = maxxi∈E

∏0≤i<j≤n

∣∣∣∣ xi − xj

1 − xixj

∣∣∣∣ .

We remark that

limn→∞

V 2/n2

n =1ρ

(see, for example, [60]). Fix 0 < q < 1. Denote by e1,n the set of points x =(x0, . . . , xn) in E × · · · × E such that

∏0≤i<j≤n

∣∣∣∣ xi − xj

1 − xixj

∣∣∣∣2

≥(

q

ρ

)n2

on e1,n . (41)

Let e2,n be the complement of e1,n in E × · · · × E. We have

E × · · · × E = e1,n ∪ e2,n

and∏

0≤i<j≤n

∣∣∣∣ xi − xj

1 − xixj

∣∣∣∣2

<

(q

ρ

)n2

on e2,n . (42)

We now estimate∫

e2,ndβn. According to (38) and (42):

∫e2,n

dβn ≤ C2(n+1)2 (n + 1)!

s0 . . . sn

∫e2,n

∏0≤i<j≤n

∣∣∣∣ xi − xj

1 − xixj

∣∣∣∣2

dμ(x0) . . . dμ(xn)

<

(q

ρ

)n2

C2(n+1)2 (n + 1)!

s0 . . . sn

∫e2,n

dμ(x0) . . . dμ(xn) .

Now, using (37), we can write

lim supn→∞

(∫e2,n

dβn

)1/n2

≤ q .


So, for any a, q < a < 1, there exists n1, such that∫e2,n

dβn ≤ an2for n ≥ n1 . (43)

Next, we prove that

I1/nn =

(∫E×···×E

e∑ n

i=0 u(xi)dβn/

∫e1,n

e∑ n

i=0 u(xi)dβn

)1/n

→ 1 , (44)

as n → ∞. We have

In = 1 +∫

e2,n

e∑ n

i=0 u(xi)dβn/

∫e1,n

e∑ n

i=0 u(xi)dβn . (45)

Remark that ∫e1,n

dβn = 1 −∫

e2,n

dβn ≥ 1 − an2for n ≥ n1 , (46)

and ∫e1,n

dβn ≤ 1 . (47)

Let M = ||u||E = maxx∈E |u(x)|. By (43), we conclude that for n ≥ n1∫e2,n

e∑ n

i=0 u(xi)dβn ≤ e(n+1)Man2

and ∫e1,n

e∑ n

i=0 u(xi)dβn ≥ e−(n+1)M

∫e1,n

dβn ≥ e−(n+1)M (1 − an2) .

Consequently, the inequalities

1 ≤ In ≤ 1 +e(n+1)Man2

e−(n+1)M (1 − an2), n ≥ n1 ,

hold, and whence (44) is true.Let us consider the integral

Jn =∫

e1,n

e∑ n

i=0 u(xi)dβn .

We have

eminxi∈e1,n(∑ n

i=0 u(xi))

∫e1,n

dβn ≤ Jn ≤ emaxxi∈e1,n(∑ n

i=0 u(xi))

∫e1,n

dβn . (48)

Since the function∑n

i=0 u(xi) is continuous on the compact set e1,n, there existspoint x∗

n = (x∗0,n, . . . , x∗

n,n) ∈ e1,n, where∑n

i=0 u(xi) attains its maximum:n∑

i=0

u(x∗i,n) = max

x∈e1,n

n∑i=0

u(xi) .


By (48) and (47),Jn ≤ e

∑ ni=0 u(x∗

i,n) .

From this and (44), we obtain

lim supn→∞

(∫E×···×E

e∑ n

i=0 u(xi)dβn

)1/n

≤ elim supn→∞( 1n

∑ ni=0 u(x∗

i,n)) . (49)

Let

σn =1n

n∑i=0

δx∗i,n

,

where δξ is Dirac’s measure supported at ξ. We have

1n

n∑i=0

u(x∗i,n) =

∫u(x)dσn(x) .

Choose a subsequence Λ ⊆ N such that

σn∗→ νq as n ∈ Λ , n → ∞ ,

and

lim supn→∞

(1n

n∑i=0

u(x∗i,n)

)=

∫u(x)dνq(x)

for a positive unit Borel measure νq and notice that supp νq ⊆ E. In view of (49),we infer

lim supn→∞

(∫E×···×E

e∑ n

i=0 u(xi)dβn

)1/n

≤ e∫

u(x)dνq(x) . (50)

By (41), for any x = (x0, . . . , xn) ∈ e1,n,

2n2

∑0≤i<j≤n

g(xi, xj) ≤1

C(E,Γ)− ln q . (51)

Let η > 0. We set

gη(ξ, t) ={

η, if g(ξ, t) > ηg(ξ, t), if g(ξ, t) ≤ η .

Remark that for all η > 0 and ξ, t ∈ G,

gη(ξ, t) ≤ g(ξ, t) . (52)

Since σn∗→ νq as n ∈ Λ, n → ∞,

σn × σn∗→ νq × νq as n ∈ Λ, n → ∞ ,

where νq × νq is a direct product of measures. For any positive Borel measure λwith support suppλ ⊆ E we define

Ig,η(λ) =∫ ∫

gη(ξ, t)dλ(ξ)dλ(t) .


Since gη(xi, xj) is continuous on E × E,

Ig,η(σn) → Ig,η(νq) as n ∈ Λ , n → ∞ . (53)

It is easy to see that

Ig,η(σn) =2n2

∑0≤i<j≤n

gη(x∗i,n, x∗

j,n) +η(n + 1)

n2.

From this, by (52), (51), and (53), we get

Ig,η(νq) ≤1

C(E,Γ)− ln q . (54)

By taking the limit η → ∞ in (54), we obtain

Ig(νq) ≤1

C(E,Γ)− ln q . (55)

We stress that νq depends on q. Letting q → 1, we conclude that thereexists a subsequence {qi}∞i=0, limi→∞ qi = 1, and a positive unit Borel measure νwith support supp ν ⊆ E such that

νqi

∗→ ν as i → ∞ .

It follows from (55) that

lim supi→∞

Ig(νqi) ≤ 1

C(E,Γ).

Since the functional Ig(ν) is lower semicontinuous,

Ig(ν) ≤ lim infi→∞

Ig(νqi) .

From this one findsIg(ν) =

1C(E,Γ)

and, by the uniqueness of the Green equilibrium measure,

ν = ω .

This implies (see (50))

lim supn→∞

(∫E×···×E

e∑ n

i=0 u(xi)dβn

)1/n

≤ e∫

u(x)dω(x) . (56)

Using a similar argument, the estimates (48), (46) and point the yn = (y0,n, . . . ,yn,n) ∈ e1,n, where

∑ni=0 u(xi) has a minimal value, one finds

lim infn→∞

(∫E×···×E

e∑ n

i=0 u(xi)dβn

)1/n

≥ e∫

u(x)dω(x) . (57)

The theorem is proved. �

Applying the arguments that are analogous to those used in the proof ofTheorem 4, we obtain immediately the following theorem.


Theorem 5. Let suppμ = [a, b] ⊂ (−1, 1) and μ′ = dμ/dx > 0 a.e. on [a, b]. Thenfor any continuous function u = u(x1, . . . , xk) : E × · · · × E → R,

limn→∞

1n

log

⎛⎝

∫E

. . .

∫E

exp

⎛⎝ ∑

i1,...,ik

u(xi1 , . . . , xik)

(n + 1)k−1

⎞⎠ dβn(x0, . . . , xn)

⎞⎠

=∫

E

. . .

∫E

u(x1, . . . , xk)dω(x1) . . . dω(xk) .

Using now the inequalities between arithmetic, geometric, and harmonicmeans, we get (for details see [16, p. 156]):

− log

⎛⎝

∫E×···×E

exp

⎛⎝−

∑i1,...,ik

u(xi1 , . . . , uik)

(n + 1)k−1

⎞⎠ dβn(x0, . . . , xn)

⎞⎠

≤∫

E×···×E

∑i1,...,ik

u(xi1 , . . . , xik)

(n + 1)k−1dβn(x0, . . . , xn)

≤ log

⎛⎝

∫E×···×E

exp

⎛⎝ ∑

i1,...,ik

u(xi1 , . . . , uik)

(n + 1)k−1

⎞⎠ dβn(x0, . . . , xn)

⎞⎠ .

Then from this and Theorem 5 we have the following result.


limn→∞

∫E

. . .

∫E

1(n + 1)k

∑i1,...,ik

u(xi1 , . . . , xik)dβn(x0, . . . , xn)

=∫

E

. . .

∫E

u(x1, . . . , xk)dω(x1) . . . dω(xk) .

In particular, for k = 1, we obtain the

limn→∞

∫E

. . .

∫E

1n + 1

(u(x0) + · · · + u(xn)

) 1(n + 1)!

(det

(Qi(xj)

))2

× dμ(x0) . . . dμ(xn)s0 . . . sn

=∫

E

u(x)dω(x) . (58)

Let

w0(x0) =∫

E

. . .

∫E

1(n + 1)!

(det

(Qi(xj)

))2 dμ(x1) . . . dμ(xn)s0 . . . sn

.

By symmetry, and by (58), we see that for any continuous function u : E → R∫E

u(x0)w0(x0)dμ(x0) →∫

E

u(x0)dω(x0) as n → ∞ ,

that isw0dμ

∗→ dω as n → ∞ .


Moreover, let

wk(x0, . . . , xk) =∫

E

. . .

∫E

1(n + 1)!

(det

(Qi(xj)

))2 dμ(xk+1) . . . dμ(xn)s0 . . . sn

.

Since the number of terms in ∑i0,...,ik

u(xi0 , . . . , xik) (59)

is (n + 1)k and the number of terms in (59) where il = ij for some l �= j is(n+1)k − (n+1) · · · (n−k+2) = O((n+1)k−1) as n → ∞, and since u is boundedon E × · · · × E, by symmetry we get from Theorem 6 the following result.


limn→∞

∫E

. . .

∫E

u(x0, . . . , xk)wk(x0, . . . , xk)dμ(x0) . . . dμ(xk)

=∫

E

. . .

∫E

u(x0, . . . , xk)dω(x0) . . . dω(xk) ,

that iswkdμ . . . dμ

∗→ dω . . . dω as n → ∞ .

5. The Friedrichs operator of a planar domain

This section contains a parallel analysis of the Hankel form given by the areameasure of a planar domain. It was Friedrichs [19] who has first investigated thisform as a mathematical tool in solving planar elasticity theory problems. He hassingled out there an important inequality for harmonic functions, in the meansquare area norm. Some of Friedrichs concepts and ideas were recently resurrectedin [49,50].

5.1. Preliminaries

Let G be a bounded domain in the complex plane C and let dA denote the areameasure in C. Denote by Lp(G, dA), 1 ≤ p ≤ ∞, the corresponding Lebesgue space.Let ALp(G), 1 ≤ p ≤ ∞, be the space of analytic functions in G which belongto Lp(G, dA). The space AL2(G) is the Bergman space. Let C : L2(G, dA) →L2(G, dA) be the complex conjugation operator: Cϕ = ϕ,ϕ ∈ L2(G, dA). TheFriedrichs operator F = FG : AL2(G) → AL2(G) of the domain G is defined as

F (ϕ) = (PC)(ϕ) , ϕ ∈ AL2(G) ,

where P is the Bergman projection, that is the orthogonal projection of L2(G, dA)onto AL2(G). Note that F is an anti-linear operator, while its square S = F 2 iscomplex linear. It is not hard to see that S is a contractive, nonnegative operatoracting on the Bergman space.


Let σ be a positive Borel measure, with closed support contained in G:suppσ = E ⊂ G. The restriction/embedding operator R : AL2(G) → L2(σ,E)is defined as before: Rϕ = ϕ,ϕ ∈ AL2(G).

In terms of forms, we have on one hand the bilinear symmetric form

[ϕ,ψ] =∫

G

ϕψdA , ϕ, ψ ∈ AL2(G) ,

and on the other hand the scalar product of the Bergman space attached to G.Friedrichs’ operator represents one by the other:

[ϕ,ψ] = 〈ϕ,Fψ〉 ,

thus, our general framework applies to T = F .

5.2. Generalized quadrature domains

We assume that the domain G admits a generalized quadrature identity supportedby the compact set E ⊂ G. This means that there exists a measure μ, suppμ = E,such that for all ϕ ∈ AL1(G)∫

G

ϕdA =∫

E

ϕ(x)dμ(x) . (60)

In particular the identity of Cauchy transforms

μ(z) = [χGdA](z) , z ∈ C \ G ,

holds. In this case we will say that G and μ are gravi-equivalent.An inverse balayage technique shows that any domain G possessing a smooth,

real analytic boundary carries such a quadrature identity. Conversely, if G admitssuch a generalized quadrature formula, then ∂G is real analytic, with a limited listof possible singular points (see for instance [52,54]).

Exactly as in the case of Hardy space, in the case when μ is a positive Borelmeasure, we can link mutatis mutandis the Friedrichs operator F to the restrictionoperator R:

〈ϕ,Fψ〉 = [ϕ,ψ] =∫

E

(ψϕ)(x)dμ(x) = 〈Rψ,CRϕ〉2,μ .

Hence we can state the following result(see also [49,50]).

Proposition 1. Let G be a planar domain which is gravi-equivalent to the positivemeasure μ, compactly supported by G. Let R : AL2(G) −→ L2(μ) be the restrictionoperator. Then the Friedrichs operator associated to G satisfies the identity:

F = R∗CR .

Example. Already known to Friedrichs, the case of an ellipse is illustrative. Specif-ically, let Ωt denote the interior of the ellipse defined by the inequality

x2

cosh2 t+

y2

sinh2 t< 1 , (61)


where t > 0 is a parameter. The basic properties of cosh and sinh, show thatas t → ∞, the domains Ωt approximate large open disks centered at the origin,while as t → 0, they are the interiors of vertically compressed ellipses, all of whichinclude the interval [−1, 1].

A conformal mapping argument (via Joukovski’s map) shows that the do-mains Ωt satisfy the generalized quadrature identity∫

Ωt

ϕ(z) dA(z) = (sinh 2t)∫ 1

−1

ϕ(x)√

1 − x2 dx , ϕ ∈ AL1(Ωt) ,

and hence the Friedrichs bilinear form [ϕ,ψ] = 〈ϕ,FΩtψ〉, ϕ, ψ ∈ AL2(Ωt), assumes

the simple representation

[ϕ,ψ] = (sinh 2t)∫ 1

−1

ϕ(x)ψ(x)√

1 − x2 dx .

In light of this, it is not surprising that we encounter Chebyshev polynomials ofthe second kind as eigenfunctions of S = F 2

Ωt. Indeed, the singular values λn(Ωt)

and normalized (in AL2(Ωt)) singular vectors ϕn are given by the formulas

λn(Ωt) =(n + 1) sinh 2tsinh[2(n + 1)t]

,

ϕn(z) =

√2n + 2

π sinh[2(n + 1)t]Un(z) ,

where Un are the Chebyshev polynomials of the second kind (see [49] or the originalpaper [19] of Friedrichs).

5.3. Eigenfunctions and eigenvalues of the Friedrichs operator

In this section we assume that G is a generalized quadrature domain, with gravi-equivalent measure μ satisfying (60). Denote by {ϕn}n the orthonormal systemof eigenfunctions of the Friedrichs form associated to a planar domain G withsmooth, real analytic boundary:

Fϕn = λnϕn .

We can choose λn non-negative, and in non-increasing order, as in Section 2.These eigenfunctions may have multiplicities, but they are all finite, due to thecompactness of F . It is not hard to see that λ0 = 1 and ϕ0 is a constant.

In the case when μ is a positive Borel measure, the integral equation satisfiedby these functions is similar to that we have encountered in the case of the Hardyspace:

λnϕn(z) =∫

G

ϕn(w)K(z, w)dμ(w) ,

where K denotes Bergman’s kernel of G. For a series of consequences of this integralequation see [29,49,50].

Theorem 1 specializes as follows.


Theorem 8. Let g0, . . . gn ∈ AL2(G). Then∣∣∣∣det(∫

G

gigjdA

)∣∣∣∣ ≤n∏

k=0

λk det(∫

G

gigjdA

). (62)

In particular, if the domain G admits a generalized quadrature identity givenby the measure μ, one obtains:

Corollary 1. ∣∣∣∣det(∫

E

(gigj)(x)dμ(x))∣∣∣∣ ≤

n∏k=0

λk det(∫

G

gigjdA

).

Next we turn to the asymptotic behavior of the eigenvalues λn.

Theorem 9. Let G be a planar domain carrying a quadrature identity given by ameasure μ supported by the compact set E ⊂ G . Then:

1.lim sup

n→∞(λ0λ1 . . . λn)1/n2 ≤ exp

(− 1/C(∂G,E)

), (63)

where C(∂G,E) is the capacity of the condenser (∂G,E);2.

lim supn→∞

λ1/nn ≤ exp

(− 1/C(∂G,E)

); (64)

3.lim infn→∞

λ1/nn ≤ exp

(− 2/C(∂G,E)

). (65)

5.4. Proof of Theorem 9

We prove only inequality (63). The other estimates (64) and (65) immediatelyfollow from (63) and the fact that the numbers λn are arranged in nonincreasingorder. Using the double orthogonality conditions (7) , we get∫

E

ϕi(x)ϕj(x)dμ(x) = δijλj ,

and, then,

λ0λ1 . . . λn = det(∫

E

(ϕiϕj)(x)dμ(x))

.

From the last relation,

(n + 1)!λ0λ1 . . . λn =∫

E

. . .

∫E

(det

(ϕi(xj)

))2

dμ(x0) . . . dμ(xn) . (66)

LetBn(x0, . . . , xn) = det

(ϕi(xj)

).

We now estimate the determinant Bn. To do this fix a positive ε > 0. We assume εsmall enough so that the set

Gε ={z : z ∈ G,dist(z, ∂G) > ε

}


is a nonempty Jordan domain containing E. Let γε be the boundary of Gε. Since∫G

|ϕi|2dA = 1 , i = 0, 1, . . . ,

we have|ϕi(z)| ≤ 1

πε2, z ∈ γε .

Using the last formula one can write

maxxi∈γε

|Bn(x0, . . . , xn)| ≤ (n + 1)!Cn+1 ,

where C = 1/πε2.Denote by g(z, ξ) the Green function of the domain Gε with singularity at

ξ ∈ Gε. Let us consider the function

Dn(x0, . . . , xn) = ln |Bn(x0, . . . , xn)|2 + 2∑

0≤i<j≤n

g(xi, xj) .

It is easy to see that Dn is a subharmonic function in Gε in each variable xi, whenthe remaining variables xj are fixed. By the maximum principle for subharmonicfunctions:

maxxi∈E

Dn(x0, . . . , xn) ≤ 2 ln((n + 1)!Cn+1

).

From this, with the help of (66), we find

λ0λ1 . . . λn ≤ (n + 1)!Cn+11 exp(−mn) , (67)

where

mn = minxi∈E

⎛⎝2

∑o≤i<j≤n

g(xi, xj)

⎞⎠ ,

and C1 is a positive constant. By well-known results of potential theory (see, forexample, [33]),

limn→∞

mn

n2=

1C(γε, E)

, (68)

where C(γε, E) is the capacity of the condenser (γε, E). By (67) and (68),

lim supn→∞

(λ0λ1 . . . λn)1/n2 ≤ exp(− 1/C(γε, E)

). (69)

By properties of the capacity, it is possible to pass to the limit on the right-handside as ε → 0, getting (63). �

Returning to the example of confocal ellipses

Ωt ={

z = x + iy;x2

cosh2 t+

y2

sinh2 t< 1

}, t > 0 ,

we remark thatlim

n→∞λn(Ωt)1/n = e−2t ,


while the modulus of Ωt (as a doubly connected domain) is exactly ρ = et. ThusC(∂Ωt, [−1, 1]) = 1/t, that is

limn→∞

λn(Ωt)1/n = exp(− 2/C

(∂Ωt, [−1, 1]

))

andlim

n→∞

(λ0(Ωt)λ1(Ωt) . . . λn(Ωt)

)1/n2

= exp(− 1/C

(∂Ωt, [−1, 1]

))

hold. In other terms, inequalities 1) and 3) in Theorem 9 are equalities for allellipses.

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Vasiliy A. ProkhorovDepartment of Mathematics and StatisticsUniversity of South AlabamaMobile, Alabama 36688-0002USAe-mail: [email protected]

Mihai PutinarDepartment of MathematicsUniversity of California at Santa BarbaraSanta Barbara, California, 93106-3080USAe-mail: [email protected]

Submitted: May 2, 2008.

Accepted: June 28, 2008.

Compact Hankel Forms on Planar Domains

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