Nonlinear Riemann-Hilbert problems with circular target curves

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Nonlinear Riemann-Hilbert Problems with Circular Target Curves

Christer Glader 1,∗ and Elias Wegert 2, ∗∗

1 Abo Akademi University, Mathematical Institute, Fanriksgatan 3B, FIN-20500 Abo, Finland2 TU Bergakademie Freiberg, Institute of Applied Analysis, D-09596 Freiberg, Germany

Received XXXX, revised XXXX, accepted XXXXPublished online XXXX

Key words Riemann-Hilbert problem, generalized modulus problem, Nehari problem, Nevanlinna-Pickinterpolation, Nevanlinna parametrization

MSC (2000) 30E25, 35Q15

The paper gives a systematic and self-contained treatment of the nonlinear Riemann-Hilbert problem withcircular target curves |w − c| = r, sometimes also called the generalized modulus problem. We assume thatc and r are Holder continuous functions on the unit circle and describe the complete set of solutions w in thedisk algebra H∞ ∩ C and in the Hardy space H∞ of bounded holomorphic functions.

The approach is based on the interplay with the Nehari problem of best approximation by bounded holo-morphic functions. It is shown that the considered problems fall into three classes (regular, singular, and void)and we give criteria which allow to classify a given problem.

For regular problems the target manifold is covered by the traces of solutions with winding number zero ina schlicht manner. Counterexamples demonstrate that this need not be so if the boundary condition is merelycontinuous.

Paying special attention to constructive aspects of the matter we show how the Nevanlinna parametrizationof the full solution set can be obtained from one particular solution of arbitrary winding number.

1 Introduction

The subject of this paper is the nonlinear Riemann-Hilbert problem with circular target curves, |w − c| = r.This problem is a special case of general Riemann-Hilbert problems with closed target curves, which have beeninvestigated in great detail (see [9], [22], [25], [27]). On the other hand, circular problems have several specificaspects, and though some have been considered in the literature (see e.g. [15] for explicit solutions, and [14]Section 3.5 for meromorphic solutions), a self-contained presentation of the subject is still missing, at leastto the best of our knowledge. The aim of this paper is to give a comprehensive treatment without assumingdifferentiability of r and c, which is required to apply the general results.

In contrast to scalar linear Riemann-Hilbert problems, which can be solved in closed form, the class consideredhere admits in general no explicit solution. Nevertheless, we tried to keep the methods as constructive as possible,especially since our investigations are also motivated by applications (see [14] p.7 and the references therein forelasticity and electrodynamics, [27] Section 6.4 for hydrodynamics, and [32] p.29 for the theory of autocorrelationequations).

Up to now nonlinear Riemann-Hilbert problems have been studied mainly under the assumption that theboundary condition is at least continuously differentiable ( [9], [22], [25], [27]). In those papers which treat lessregular problems ( [2], [20]) only the existence of bounded solutions has been shown and no (sufficiently weak)conditions were given which guarantee continuity of solutions on the closed disk. Here we close this gap usingan approach based on an extremal property of the solutions which relates the boundary value problem to the

∗ Supported by the Research Institute of the Abo Akademi University Foundation and the Magnus Ehrnrooth Foundation.∗∗ Corresponding author E-mail: [email protected], Phone: +49 3731 39 2689, Fax: +49 3731 39 3442,

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2 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

Nehari problem of best approximation by holomorphic functions. We preferred to do this without recourse toAdamyan-Arov-Krein theory [1], though several results could also be obtained this way.

For the convenience of the reader we start with summarizing some facts about linear problems. Using theinterplay between the Riemann-Hilbert problem and the Nehari problem, we infer the existence of a continuoussolution from the Carleson-Jacobs theorem. In order to obtain all solutions we then apply simple transformationswhich reduce the problem to a linear one. Moreover we explicitly construct the Nevanlinna parametrization ofthe full solution set. As a by-product we get a simple proof for uniqueness of this parametrization for the classicalNevanlinna-Pick interpolation problem.

Since the coefficients of the Nevanlinna parametrization are expressed in terms of the solution to the Nehariproblem and the Hilbert transform, we have complete control on their regularity. This gives a convenient approachto study the regularity of solutions to Riemann-Hilbert problems not only in spaces of bounded or (Holder)continuous functions as we do it here, but in many other relevant spaces as well, for example in the “decentspaces” introduced in Peller’s book [17].

In its classical setting, the Riemann-Hilbert problem (as we understand it here) consists in finding all functionsw in the disk algebra H∞ ∩ C, i.e. the Banach algebra of all holomorphic functions in the complex unit disk Dwhich extend continuously onto the closed disk D, satisfying the boundary condition

F (t, w(t)) = 0

on the unit circle T. Here F : T× C → R is a given real-valued function.

The problem dates back to Bernhard Riemann’s famous thesis in 1851. The reader interested in its historymay consult [27], [28]. Riemann himself considered the problem from a geometric point of view. Following him,we introduce the target curves

Mt := {z ∈ C : F (t, z) = 0}, t ∈ T,

and rewrite the boundary condition as w(t) ∈ Mt. The target curves can be considered as fibers of the targetmanifold

M := {(t, z) ∈ T× C : F (t, z) = 0},

and defining the trace of a solution (or, more generally, of a complex valued function on T or on a supersetthereof) by

trw := {(t, w(t)) ∈ T× C : t ∈ T},

the boundary condition attains the simple form trw ⊂M .Depending on whether the target curves are open or closed we distinguish two classes of Riemann-Hilbert

problems with different properties. Since we shall consider problems with circular target curves, only the secondclass is relevant here. For smooth target manifolds of class C2 “regular problems” have been first investigated infull generality by Alexander Shnirel’man [22] using a topological degree for quasi-linearlike mappings introducedby him. Using “von Wolfersdorf’s trick” ( [31]) and the Leray-Schauder degree Wegert [25] extended the resultsto (compact and non-compact) target manifolds of class C1 and described the different properties of the threeclasses with compact target manifold. An independent approach, based on the implicit function theorem, wasdeveloped by Forstneric [9].

Meanwhile several results are also available for Riemann-Hilbert problems with piecewise continuous bound-ary conditions ( [12], [21]). Riemann-Hilbert problems on multiply connected domains and on bordered Riemannsurfaces were studied by Efendiev and Wendland [7], [8], and Cerne [5], [6] respectively. Far reaching generaliza-tions of nonlinear Riemann-Hilbert problems appear in investigations of analytic disks with boundaries attachedto a submanifold of Cn and in H∞-optimization (see Whittlesey [29], [30] and the references therein).

In order to formulate the existence result we need some terminology. Any solution w ∈ H∞ ∩ C possessesa topological characteristic, its winding number about the target manifold. To define this number windMw we

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fix any continuous function m on T such that m(t) belongs to the interior intMt of the target curve Mt for allt ∈ T, which we abbreviate as trm ⊂ intM . Then

windMw := wind (w −m),

where the “wind” on the right refers to the usual winding number of a (nonvanishing continuous complex–valued) function on T about the origin. The winding number about M does not depend on the choice of m, and if0 ∈ intMt for all t ∈ T, then windMw coincides with the number of zeros of w in D, counted with multiplicity.

A compact (target) manifold M ⊂ T× C is said to be admissible, if it has a parametric representation

M = {(t, µ(t, s)) : t, s ∈ T}

with a continuously differentiable function µ on T2, such that the derivative ∂sµ does not vanish on T2 and themappings T →Mt, s 7→ µ(t, s) are injective for all t ∈ T.

It turned out (see [26], [27]) that the class of admissible compact target manifolds splits into three subclasseswhich we define next.

An (admissible) compact target manifold M is said to be regularly traceable if there exists a holomorphicfunction m ∈ H∞ ∩ C such that trm ⊂ intM , and any such function m is called a center of M . If M is notregularly traceable but there still exists a holomorphic function m ∈ H∞ ∩ C such that trm ⊂ clos intM , it issaid to be singularly traceable. If M is neither regularly nor singularly traceable we call it nontraceable or void.To simplify terminology we shall also speak of regular, singular, and void Riemann-Hilbert problems. Criteriafor classifying a specific problem will be given later (see Lemma 6.1 and Theorems 7.1–7.4).Here is the basic result about regular Riemann-Hilbertproblems with differentiable compact target manifold:Theorem 1.1 ( [22], [25], [27] Theorem 2.5.1)Any regularly traceable compact target manifold Mof class C1 is covered by the traces of solutions tothe Riemann-Hilbert problem with winding number zeroabout M in a schlicht manner.

Figure 1 shows a compact target manifold M andsome traces of solutions with winding number zero.Together with the corresponding (liftings of the) tar-get curves these lines define a natural (“holomorphic”)parametrization of M . Figure 1

In the present paper we consider the special class of problems with circular target curves,

Mt := {(t, w) ∈ T× C : |w − c(t)| = r(t)},

with a complex valued function c and a strictly positive function r. If the functions c and r are continuouslydifferentiable the corresponding target manifold is admissible and Theorem 1.1 applies. Here we shall provethat the result extends to the case where c and r are Holder continuous. Note that assertion (i) of the followingtheorem does in general not hold if r and c are merely continuous (see the counterexamples of Lemma 8.1 andLemma 8.2).

Theorem 1.2 Let r, c ∈ Cα(T) with 0 < α < 1 and r > 0, and assume that the Riemann-Hilbert problem|w − c| = r is regular, i.e., there exists a function m ∈ H∞ ∩ C such that |m(t) − c(t)| < r(t) for all t ∈ T.Then the following assertions hold:

(i) The target manifold M is covered by the traces of solutions w in H∞ ∩ C with winding number zero aboutM in a schlicht manner.

(ii) Let n be a nonnegative integer. Then, for all t0 ∈ T, z0 ∈ C with |z0 − c(t0)| = r(t0), and z1, . . . , zn ∈ D,there exists a unique solutionw ∈ H∞∩C with winding number n aboutM which satisfies the interpolationconditions

w(t0) = z0, and w(zj) = m(zj) for j = 1, . . . , n. (1)

4 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

(iii) Assertion (ii) describes the complete set of solutions in H∞ ∩ C and any such solution extends on D to afunction in the Holder space Cα(D).

If there are multiple interpolation nodes zj , the interpolation conditions (1) must be modified accordingly.

2 Spaces and Transformations

For 1 ≤ p ≤ ∞ we denote by Hp the familiar holomorphic Hardy spaces. All spaces we shall be concerned withare subspaces of H1, such that the functions admit nontangential limits almost everywhere on T and values in Dcan be recovered from the boundary function by Cauchy’s (or Poisson’s) integral formula. Frequently we shallidentify holomorphic functions with their boundary function, and we shall speak of a holomorphic function on Tif it extends holomorphically into D.

The regularity of holomorphic functions in D is also measured in spaces of the corresponding boundary func-tions. In particular we denote byH∞∩Ck+α (k ∈ Z+, 0 < α < 1) the spaces of bounded holomorphic functionsin D with boundary function in the Holder class Ck+α(T), and by H∞ ∩W k

p (k ∈ Z, k ≥ 1, 1 ≤ p < ∞) theHardy-Sobolev spaces with boundary functions in the Sobolev space W k

p (T). All these spaces are algebras. Notethat H∞ ∩ Ck+α consists of all holomorphic functions functions in Ck+α(D).

In what follows we need transformations of target manifolds and solutions. For a given function T : T×C →C we define

T (M) := {(t, T (t, z)) : (t, z) ∈M}.

In order to guarantee that a transformation T (M) of a target manifold M is again a target manifold the functionT must satisfy additional conditions. For circular Riemann-Hilbert problems a natural class of transformations isgenerated by fibered Mobius transforms

T (t, z) :=a(t) z + b(t)c(t) z + d(t)

with complex valued continuous functions a, b, c, d on T. We shall consider such transformations in Section 5.For the time being we only need linear transformations

M = aM + b := {(t, a(t)z + b(t)) : (t, z) ∈M}.

If, in particular, a and b are in H∞ ∩ C and a has no zeros on T, then any solution w of trw ⊂ M generates asolution w := aw + b of trw ⊂ M and

windfM w = windM w + wind a.

Two Riemann-Hilbert problems with target manifolds M and M are said to be linearly equivalent, if there existfunctions a and b in H∞ ∩ Cα with wind a = 0. The solutions of linearly equivalent problems are in a bijectiverelation via w = aw + b.

Lemma 2.1 Every Riemann-Hilbert problem |w − c| = r with c, r ∈ Cα(T) and r > 0 is linearly equivalentto a problem |w − c | = 1 with c ∈ Cα(T).

P r o o f. Since r is strictly positive and Holder continuous, also log r belongs to Cα(T). Hence the functionw0 defined by the Schwarz integral

w0(z) := exp12π

∫T

t+ z

t− zlog r(t) |dt|, z ∈ D, (2)

is holomorphic in D and extends continuously to a function in Cα(D) with |w0(t)| = r(t) for all t ∈ T. Becausethe Holder spaces Cα(T) are algebras and w0 has no zeros in D, the problem |w − c| = r is equivalent to|w − c | = 1 with c := c/w0.

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If the target circles are concentric, i.e. c = const, the problem |w − c| = r can be solved explicitly. Namely,the shift M 7→ M − c and the above transformation lead to the linearly equivalent problem |w| = 1. Thecontinuous solutions of the latter are exactly the finite Blaschke products

B(z) = c

n∏j=1

zj − z

1− zjz,

where c ∈ T and zj ∈ D can be chosen arbitrarily.

3 Linear Riemann-Hilbert Problems

The target curves of linear Riemann-Hilbert problems are straight lines. Traditionally the boundary condition iswritten in the form

Im(f(t)w(t)

)= g(t) (3)

and the function f is said to be the symbol of the problem. Note that (3) fits with the general setting of Riemann-Hilbert problems with nonlinear boundary condition F (t, w(t)) = 0 if we define

F (t, z) := Im(f(t) z

)− g(t), t ∈ T, z ∈ C.

In the sequel we encounter problems with continuous non-vanishing symbols. Since then, in general, continuoussolutions do not exist, we admit solutions in the Hardy spaces Hp (1 < p <∞), which fits with right-hand sidesg in the Lebesgue spaces Lp

R(T), and the boundary condition is required to hold almost everywhere on T.The winding number κ := wind f of f is also called the index of problem (3). The following proposition

collects relevant results for problems with index zero, where, after dividing the boundary condition by |f |, thesymbol f can be written as f = exp(iϕ) with a continuous function ϕ on T. The explicit solution formulainvolves the Hilbert operator, defined by the principle value integral

Hϕ(ei τ ) =12π

∫ 2π

0

ϕ(ei σ) cotσ − τ

2dσ.

Proposition 3.1 Let f = exp(iϕ) with a real-valued continuous function ϕ on C(T).

(i) If g ∈ Lp with 1 < p <∞ the general solution of (3) in Hp is given by

w = wg + d · w0,

where d is an arbitrary real constant and

w0 := exp(Hϕ+ iϕ), wg := w0 ·(H (g/|w0|) + i (g/|w0|)

).

The operator Lp(T) → Hp, g 7→ wg is continuous.

(ii) If f and g are in Ck+α(T) with k ∈ Z, k ≥ 0, and 0 < α < 1, then w ∈ H∞ ∩ Ck+α.

(iii) If f and g are in W kp (T) with k ∈ Z, k ≥ 1, and 1 < p <∞, then w ∈ H∞ ∩W k

p .

The first result is known to specialists for nearly fifty years and is by now part of mathematical folklore.References to the precise statements are nevertheless hard to find since they frequently appear in the language ofsingular integral equations (see Gohberg and Krupnik [11]) or Toeplitz operators (see Bottcher and Silbermann[3], Theorem 2.42). A straightforward proof of all assertions is in Section 1.7 of [27].

Linear Riemann-Hilbert problems with arbitrary index κ can easily be reduced to the case κ = 0. We denoteby ϕ any branch of arg f which is continuous on T \ {1}, set

ϕ0 :=12π

∫Tϕ(t) |dt|, f0 := exp(iϕ0), (4)

and define the continuous function ψ on T by

ψ(eiτ ) := ϕ(eiτ )− κτ, 0 ≤ τ < 2π. (5)

6 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

Proposition 3.2 Let 1 < p < ∞, f ∈ C(T), |f | ≡ 1 on T, and let f0 and ψ be given by (4) and (5),respectively. Then the following assertions hold:

(i) The Riemann-Hilbert operator

Rf : Hp → LpR, w 7→ Im (fw)

is Fredholm and satisfies

dim kerRf = max{0, 2κ+ 1}, codim imRf = max{0,−2κ− 1}.

(ii) Let κ ≥ 0, fix κ pairwise distinct points z1, . . . , zκ ∈ D \ {0} and set p0 :=∏zj . Then, for arbitrarily

chosenw1, . . . , wκ ∈ C, γ ∈ R, and δ ∈ T with Im(δ p0f0

)6= 0, the Riemann-Hilbert problem Im ( fw) =

g admits a unique solution w ∈ Hp which satisfies the interpolation conditions

Im ( δ w(0)) = γ, w(zj) = wj (j = 1, . . . , κ).

(iii) If κ < 0 the problem Im ( fw) = g is solvable if and only if the 2|κ| − 1 conditions∫ 2π

0

g(eiτ ) exp(−Hψ(eiτ )) cos kτ dτ = 0 (k = 0, 1, . . . , |κ| − 1) (6)∫ 2π

0

g(eiτ ) exp(−Hψ(eiτ )) sin kτ dτ = 0 (k = 1, . . . , |κ| − 1) (7)

are satisfied. If a solution exists it is unique.

P r o o f. 1. Note that the functionals defined in (6) and (7) are continuous in Lp, since the continuity of ψimplies that exp(−Hψ) belongs to Lp(T) for all p < ∞ (see, for instance Corr. 2.6 in Sec. III.2 of [10]). Thefirst assertion (i) then follows immediately from (ii) and (iii).

2. Let κ = 0. Since the function Hϕ + iϕ is holomorphic and has vanishing real part at z = 0 we havew0(0) = exp(iϕ0) = f0, where ϕ0 denotes the mean value of ϕ over T. By Proposition 3.1 the general solutionis w = wg + dw0 and because Im

(δf0

)6= 0 there is exactly one real constant d such that Im

[δ(wg(0) +

dw0(0))]

= γ.

3. If κ > 0 let p(z) :=∏κ

j=1(z − zj) and let q be a polynomial satisfying the interpolation conditionsq(zj) = wj . The functionw is a solution of the Riemann-Hilbert problem Im ( fw) = g and satisfiesw(zj) = wj

if and only if w = pw + q, where w is a solution of the Riemann-Hilbert problem

Im(f p w

)= g − Im ( f q), (8)

which has index zero. The function ϕ defined by

ϕ(t) := arg(f(t)p(t)

)= ϕ(t)−

∑arg(t− zj)

is continuous on T and its mean value (modulo 2π) is ψ0 = ϕ0 − κπ. Consequently the solution w0 of thehomogeneous problem (8) satisfies w0(0) = (−1)κ exp(iϕ0) = (−1)κf0. The assumption Im

[δ p0f0

]6= 0

then guarantees that there is exactly one solution w of the inhomogeneous problem (8) such that Im(δ w(0)

)=

Im(δ (p(0)w(0) + q(0))

)= γ.

4. If κ < 0 any solution w of (3) generates a solution w(z) := z|κ| w(z) of the Riemann-Hilbert problemIm

(ftκ w(t)

)= g(t) with index zero. The general solution of this problem is

w = wg + dw0 = w0

(d+H(g/|w0|) + i g/|w0|

).

A solution of the original problem exists if and only if w has a zero of order |κ| at z = 0, and since w0(0) =exp

(iψ(0)

)6= 0 the (holomorphic extension of the) factor d + H(g/|w0|) + i g/|w0| must vanish of order |κ|.

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This happens if and only if the Fourier coefficients of the boundary function vanish up to order |κ| − 1. Takinginto account the mapping properties of H in Fourier space we get d = 0 and the conditions for the imaginary part∫

T

g(t)|w0(t)|

tk |dt| = 0, k = 0, . . . , |κ| − 1.

With |w0| ≡ exp(Hψ) assertion (iii) follows.

We close this section with some specific results which are needed in the following.

Lemma 3.3 Let f be a continuous non-vanishing function on T.

(i) If wind f = 0 and w ∈ Hp is a nontrivial solution of the homogeneous problem Im(f w

)= 0 with p > 1,

then the function Re(f w

)is either positive almost everywhere on T or negative almost everywhere on T.

(ii) If f ∈ Cα(T) and wind f ≥ 0 there exists w ∈ H∞ ∩ Cα such that Im(f w

)= 0 and Re

(f w

)> 0 on

T.

P r o o f. (i) After unimodular normalization f = exp(iϕ) we have Re(fw

)= d exp(Hϕ) with d 6= 0. Since

exp(Hϕ) ≥ 0, and exp(±Hϕ) ∈ Lp(T) (see the proof of Proposition 3.2), this implies that exp(Hϕ) ∈ R+

almost everywhere on T.

(ii) If ϕ is Holder continuous on T then exp(Hϕ) > 0. The assertion for wind f = 0 then follows from therepresentation of w0 in Lemma 3.3. If κ = wind f > 0 it suffices to apply this result with f(t) replaced byt−κf(t).

The next lemma is a statement about the location of the value w(0) for holomorphic functions with boundaryvalues restricted to (rotating) half-planes.

Lemma 3.4 Let f ∈ C(T) with f 6= 0 on T and assume that w ∈ Hp with p > 1 satisfies Im(f w

)≥ 0

almost everywhere on T.

(i) If wind f = 0 then Im(f0 w(0)

)≥ 0, where f0 is given by (4).

(ii) If wind f < 0 then w ≡ 0.

P r o o f. 1. The nontrivial solution w0 of the homogeneous problem Im(f w

)= 0 and its inverse 1/w0

belong to Hp for all p <∞. Hence w/w0 is in H1 and since Im(w/w0

)≥ 0 almost everywhere on T this also

holds at all points z ∈ D. For z = 0 this is equivalent to assertion (i).

2. If κ := wind f < 0, the function w defined by w(z) = z|κ| w(z) satisfies the boundary conditionIm

(t−κf(t) w(t)

)≥ 0. The function t 7→ t−κ f(t) has winding number zero and hence the corresponding

linear Riemann-Hilbert problem Im(t−κf(t)w(t)

)= 0 has a nontrivial solution w0. Now the first step shows

that Im(w/w0

)≥ 0 in D, and since w0(0) 6= 0, the values w(z) lie in a certain sector with opening angle less

than 2π, for all z in a sufficiently small neighborhood of 0. By the open mapping principle for holomorphicfunctions this is only possible if w is constant, i.e. w ≡ w(0) = 0, and hence w ≡ 0.

4 Existence of Solutions

In this section we prove the main existence theorems for solutions of regular and singular Riemann-Hilbert prob-lems |w− c| = r with Holder continuous functions c and r. In both cases we use classical results about solutionsof the Nehari problem of best uniform approximation by bounded holomorphic functions. For convenient refer-ence we recall some facts.

Proposition 4.1 (i) Each f in L∞(T) has a best approximation w0 in H∞,

‖f − w0‖∞ = infw∈H∞

‖f − w‖∞.

8 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

(ii) If f ∈ C(T) the function w0 is unique and the error f − w0 has constant modulus,

|f(t)− w0(t)| = const a.e. on T. (9)

(iii) If f is Dini continuous, i.e. its modulus of continuity ω satisfies∫ 1

0

ω(x)x

dx <∞,

then its best approximation w0 is continuous.

(iv) If f and w0 are continuous the winding number of f − w0 about zero is negative.

(v) If f is in Cα(T) for some 0 < α < 1, then w0 belongs to H∞ ∩ Cα.

Assertions (i) and (ii) follow from Garnett [10] Theorem 4.1.3 and Theorem 4.1.7, or the material on pp. 36–40of Partington’s book [16]. The statements (iii) and (v) are in Theorem 3 of Carleson’s and Jacobs’ paper [4] (seealso [10] Theorem 4.2.1 for (iii), and Peller [17] Theorem 2.4 in Section 7.2 and Section 7.4 for (v)). Assertions(ii) and (iv) together are sometimes called “Poreda’s criterion”, see Poreda [18].

A short proof of (iv) is as follows. Assume that wind (w0 − f) ≥ 0, approximate f − w0 by a smoothfunction g 6= 0 such that |arg g − arg (w0 − f)| < π/2. Then the index of the Riemann-Hilbert problemIm (g w) = 0 is nonnegative and, by Lemma 3.3 (ii), it has a Holder continuous solution w1 with Re (g w1) > 0on T. Consequently |w0 + λw1 − f | < |w0 − f | for all sufficiently small positive λ, which is in conflict with theoptimality of w0.

Using these results the existence of a Holder continuous solution to singular problems is easy to prove.

Theorem 4.2 Let r, c ∈ Cα(T), with 0 < α < 1 and r > 0, and assume that the Riemann-Hilbert problem|w−c| = r is singular. Then the problem has exactly one solution inH∞∩C. This solution belongs toH∞∩Cα

and has a negative winding number about M .

P r o o f. According to Lemma 2.1 we can assume that r ≡ 1. Let w0 be the best approximation of c in H∞.Then w0 ∈ H∞ ∩Cα and |w0(t)− c(t)| = const =: λ on T. Clearly λ ≤ 1 since ‖m− c‖∞ ≤ 1. If λ were lessthan 1 the problem would be regular. So λ = 1 and w0 is a solution with windMw0 = wind (w0 − c) < 0.

Since any other solution of the Riemann-Hilbert problem would produce another best approximation of c,uniqueness follows from (ii) in Proposition 4.1.

To prove the main existence result for regular problems we consider the set A of bounded holomorphic func-tions with restricted boundary values

A := {w ∈ H∞ : |w(t)− c(t)| ≤ r(t) a.e. on T} (10)

and study extremal properties of functions in A. We start with some technical results. The first is quoted fromLemma 2.3.1 in [27].

Lemma 4.3 LetG0 andG1 be bounded Jordan domains satisfying closG0 ⊂ G1, and let t0 ∈ T, δ0 > 0, andC0 > 0 be fixed. Then there exists a positive number δ with the following property: ifw ∈ H∞, ‖w‖∞ ≤ C0, andw(t) ∈ G0 almost everywhere on {t ∈ T : |t− t0| < δ0}, then w(z) ∈ G1 for all z ∈ {z ∈ D : |z − t0| < δ}.

In the next step we investigate the location of w(z) for functions w in A with z close to t ∈ T. Here it issufficient to assume that r and c are continuous.

Lemma 4.4 Let r, c ∈ C(T). Then for each ε > 0 there exists a δ > 0 such that the following implicationholds:

w ∈ A, z ∈ D, t ∈ T, |z − t| < δ ⇒ |w(z)− c(t)| < r(t) + ε.

P r o o f. Fix t0 ∈ T. By Lemma 4.3 there exists a number δ1 > 0 such that

w ∈ A, z ∈ D, |z − t0| < δ1 ⇒ |w(z)− c(t0)| < r(t0) + ε/2.

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Using the continuity of r and c we get the existence of a positive number δ2 such that

w ∈ A, z ∈ D, t ∈ T, |z − t0| < δ1, |t− t0| < δ2 ⇒ |w(z)− c(t)| < r(t) + ε. (11)

Varying t0 over T we find the corresponding numbers δ1(t0) and δ2(t0) for each point t0 on T. We define δ3(t) :=12 min(δ1(t), δ2(t)). The compactness of T ensures the existence of an annulus {z ∈ C : 1−δ < |z| < 1} whichis covered by a finite collection of open sets {z ∈ D : |z − tk| < δ3(tk)} with tk ∈ T and k = 1, . . . , n. Wemay assume that δ < min δ3(tk). For arbitrarily given z ∈ D and t ∈ T with |z − t| < δ there exists a numbertk such that

|z − tk| < δ3(tk) < δ1(tk),

and then we have

|t− tk| ≤ |t− z|+ |z − tk| < δ + δ3(tk) ≤ δ2(tk).

These two relations together show that |w(z)− c(t)| < r(t) + ε for all w ∈ A (cp. (11)).

Recall that normal convergence of a sequence of functions defined in D means uniform convergence on everycompact subset of D.

Lemma 4.5 If c and r are continuous the set A is (empty or) compact with respect to normal convergence.

P r o o f. 1. Assume that A is not empty. Since A is a uniformly bounded family of holomorphic functions itis relatively compact with respect to normal convergence by Montel’s theorem.

2. In order to prove that A is closed let (wn) ⊂ A be a sequence which converges normally to a function w.Since the limit function w is holomorphic in D and A is a bounded subset of H∞ we have w ∈ H∞.

Let ε be an arbitrarily given positive number. Choosing δ according to Lemma 4.4, we conclude that

|wn(λt)− c(t)| < r(t) + ε for all t ∈ T, n ∈ Z+, 1− δ < λ < 1.

Taking first the limit n→∞ and then λ→ 1− 0 we see that

|w(t)− c(t)| ≤ r(t) + ε

for any point t ∈ T where the nontangential limit of w exists. Since ε > 0 was chosen arbitrarily w ∈ Afollows.

Remark. A similar result holds for sets of holomorphic functions with more general restrictions of the bound-ary values (see Semmler [20], Lemma 2.3.6).

After these preparations we are in a position to show that regular Riemann-Hilbert problems have solutions.Lemma 4.6 Let c, r ∈ Cα(T). If the set A defined by (10) is not empty, then the Riemann-Hilbert problem

|w − c| = r has a solution w in H∞ ∩ Cα with windMw ≤ 0.

P r o o f. By Lemma 2.1 it suffices to consider the case where r ≡ 1. We introduce the range of values offunctions in A at z = 0 as the set

Y := {w(0) : w ∈ A}.

Since A is not empty and compact the same holds for Y . In particular the boundary ∂Y of Y is not empty andwe pick a point W0 ∈ ∂Y . The function c defined by

c(t) := t−1 (c(t)−W0)

is in Cα(T) and has a (unique) best approximation w0 in H∞ which belongs to H∞ ∩ Cα (see Proposition 4.1(i) and (v)). By Proposition 4.1 (ii) the function w0 defined by w0(z) = z w0(z) +W0 satisfies

|w0(t)− c(t)| = |w0 − c(t)| = const =: λ

10 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

for all t ∈ T. In order to prove that λ = 1 we first choose a function w ∈ A with w(0) = W0 (which exists sinceW0 ∈ ∂Y ⊂ Y ) and define w by

w(z) := z−1 (w(z)−W0).

Then w is inH∞ and satisfies |w− c | = |w−c| ≤ 1 almost everywhere on T, which yields that λ ≤ 1. If λ wereless than one, then w0 + d would belong to A for all d ∈ C with sufficiently small modulus. This would implythatw0(0) is an interior point of Y , which is a contradiction sincew0(0) = W0 ∈ ∂Y . Finally, by Proposition 4.1(iv), the winding number of w0 − c is negative and hence

windMw0 = wind (w0 − c) = 1 + wind (w0 − c) ≤ 0.

If the problem is regular all solution must have a nonnegative winding number about M . Indeed, if m ∈H∞ ∩ C is a function with |m− c| < r the argument principle implies that windMw0 = wind (w0 −m) ≥ 0.

Once the existence of one solution w0 ∈ H∞ ∩ Cα with winding number zero has been shown we can give aconstructive proof of assertion (i) in Theorem 1.2.

The transformation M 7→ M := 1/(M − w0) sends the target curves Mt to straight lines Mt. The resultinglinear Riemann-Hilbert problem has Holder continuous coefficients and index ind M = 0. By Proposition 3.1the solutions are Holder continuous and their traces cover the target manifold M in a schlicht manner.

Since 0 /∈ Mt for all t ∈ T, the winding number of any solution w about zero is equal to ind M , i.e. zero,which implies that the solutions have no zeros in D. Consequently, the functions w, obtained from the solutionsw by the transformation w := w0 + 1/w, are holomorphic in D and Holder continuous on D. Their traces coverM \ trw0, which together with the trace of w0 yields a schlicht covering of M . We parameterize this family ofsolutions ws by s ∈ T.

To complete the proof of assertions (i) and (iii) in Theorem 1.2 for solutions with winding number zero itremains to show that any continuous solution w of trw ⊂M with winding number zero about M is equal to ws

for some s ∈ T.Let s ∈ T be arbitrarily chosen. Considering the straight lines Lt through the points w(t) and ws(t), which

are replaced by the tangents to Mt at ws(t) if w(t) = ws(t), we see that w and ws are both solutions of a linearRiemann-Hilbert problem with continuous (but not necessarily Holder continuous) coefficients and index zero.By Lemma 3.3 the traces of w and ws cannot “cross” each other. Since this must hold for all solutions ws, andsince the traces of ws cover all of M , w must coincide with one of the functions ws.

Remark. Explicit computation of the solutions ws needs two applications of the Hilbert operator. If eventwo solutions w1 and w2 with winding number zero are known the construction can be simplified, since then therational transformation

M 7→ M − w1

M − w2

maps the target curves Mt to straight lines through the origin, and solving the resulting homogeneous linearRiemann-Hilbert problem requires only one application of the Hilbert operator.

The statements of Theorem 1.2 about solutions with positive winding number n are reduced to the case n = 0.In order to do so, we set q(z) :=

∏nj=1(z − zj) and define the transformed target manifold M := (M −m)/q.

If w and w are related by w = m + q w, then w is a solution of the Riemann-Hilbert problem trw ⊂ M andsatisfies (1) if and only if w solves trw ⊂ M . In this case the winding numbers of w and w are related by

windM w = windfM w + n.

Since the transformed target manifold is also regular the proof of Theorem 1.2 is complete.

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5 Nevanlinna Parametrization

In this section we show how the complete set of solutions to |w − c| = r can be obtained from three solutionswith winding number zero. This yields a natural parametrization of all solutions which goes with the nameNevanlinna parametrization. Though the result could be derived from the classical case, where r ≡ 1, usingthe transformation to the normalized form in Lemma 2.1, we give an independent and more direct proof. As asurplus we immediately get uniqueness of the parametrization up to a conformal automorphism of the unit disk(see Takahashi [23], [24] for the generalized Nevanlinna-Pick problem).

A Nevanlinna parametrization N is a fibered Mobius transform defined on the set of finite Blaschke products(or, more generally, on the set of unimodularly bounded functions),

(NB)(z) := Nz(B(z)), z ∈ D, (12)

where all fiber maps Nz are Mobius transforms restricted to the closed disk D,

Nz(b) :=P (z) b+Q(z)R(z) b+ S(z)

, b, z ∈ D. (13)

Theorem 5.1 Let r, c ∈ Cα(T) with 0 < α < 1 and r > 0, and assume that the Riemann-Hilbert problem|w − c| = r is regular.

(i) There exist holomorphic functions P,Q,R, S ∈ H∞ ∩ Cα such that the fibered Mobius transform N givenby (12), (13) is a bijective mapping between the set B of finite Blaschke products and the solution set W ofthe Riemann-Hilbert problem |w − c| = r.

(ii) IfN1 andN2 are two such parameterizations, then there exists a conformal automorphism g of the unit disksuch that N2 = N1 ◦ g, i.e.

N2B(z) = N1

(g(B(z))

)for all Blaschke products B and all z ∈ D.

(iii) The coefficients of any such parametrization satisfy

|R(z)| < |S(z)|, P (z)S(z) 6= Q(z)R(z), z ∈ D.

Note that the Nevanlinna parametrization can be normalized to S ≡ 1 since S has no zeros in D,

P r o o f. 1. Since the problem is regular it has three solutions w1, w2, w3 with winding number zero about M .We choose the numbering so that w1(t), w2(t), w3(t) are positively oriented on the circle Mt for t = 1. Sincethe traces of different solutions with winding number zero about M do not intersect each other, the three pointshave the same orientation for all t ∈ T.

Because the winding number of the boundary function of wj − wk (with j 6= k) about the origin is zero, itfollows from the argument principle that for all z ∈ D the three points w1(z), w2(z), w3(z) are pairwise distinct.So the function w := (w3 − w2)/(w2 − w1) is holomorphic in D and, due to the orientation of the pointsw1(t), w2(t), w3(t), it has positive imaginary part on T. This implies that Imw(z) is positive for all z ∈ D, fromwhich we conclude that w1(z), w2(z), w3(z) lie on a (proper) circle Mz and have positive orientation.

2. In order to construct a Nevanlinna parametrization we fix z and consider the (unique) Mobius transformNz which maps the third roots of unity ω1, ω2, ω3 to the values w1(z), w2(z), w3(z) preserving orientation.Since the transform Nz maps (the interior of) T to (the interior of) the circle Mz a well-known result aboutMobius transforms tell us that the coefficientsP (z), Q(z), R(z), S(z) satisfy (iii). These coefficients are uniquelydetermined up to a common factor and can be easily computed from the invariance of the cross ratio,

w − w2

w − w3÷ w1 − w2

w1 − w3=b− ω2

b− ω3÷ ω1 − ω2

ω1 − ω3,

12 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

which results in

P = ω1 w1 (w3 − w2) + ω2 w2 (w1 − w3) + ω3 w3 (w2 − w1)

Q = ω1ω2 w3 (w2 − w1) + ω2ω3 w1 (w3 − w2) + ω3ω1 w2 (w1 − w3)

R = ω1 (w3 − w2) + ω2 (w1 − w3) + ω3 (w2 − w1)

S = ω1ω2 (w2 − w1) + ω2ω3 (w3 − w2) + ω3ω1 (w1 − w3).

This shows that P,Q,R and S are holomorphic in D and have the same regularity as the solutions w1, w2, w3.

3. For t ∈ T the Mobius transformNt maps T to the target circleMt. Consequently, for any Blaschke productB, the function w = NB defined by

NB(z) :=P (z)B(z) +Q(z)R(z)B(z) + S(z)

, z ∈ D

is a solution of the Riemann-Hilbert problem. In order to find the winding number of w about M we remark thatthe function m := Q/S satisfies m(t) ∈ intMt for all t ∈ T. Since

w −m = BSP −RQ

S(S +RB)

and, by (iii), S, S +RB, SP −RQ have no zeros in D, we obtain that windMw = degB.4. If B is Blaschke product of degree n with zeros at z1, . . . , zn thenNB is a solution of the Riemann-Hilbert

problem with winding number n about M and w(zj) = m(zj). Moreover, for any t0 ∈ T the mapping

b 7→ w(t0) =P (t0)b+Q(t0)R(t0)b+ S(t0)

is a bijection of T onto Mt0 . This again proves the existence part of assertion (ii) in Theorem 1.2 and, togetherwith the uniqueness statement (iii) of Theorem 1.2, it follows that N maps the set of finite Blaschke productssurjectively onto the solutions of the Riemann-Hilbert problem. It is clear that N is injective.

5. Any Nevanlinna parametrization N has the form (12), (13). Since all Mobius transforms Nz must map Dto proper disks (see above), the functions S, S +RB, and SP −RQ have no zeros in D.

Hence, according to the third step, the constant functions must be mapped to solutions with winding numberzero about M and vice versa. If w1, w2, w3 are three such solutions, the parametrization is uniquely defined oncethe three unimodular constants (functions) νj := N−1wj , j = 1, 2, 3, are chosen (with the correct orientation onT). If N1 denotes the standard parametrization from above and g is the (unique) conformal automorphism of Dwhich sends νj to ωj , then N and N1 ◦ g are both Nevanlinna parameterizations which map νj to wj and hencethey must coincide.

The Nevanlinna parametrization is a convenient tool to study (generalized) solutions in the Hardy spaces Hp,where the boundary values w(t) are to be understood in the sense of nontangential limits and the boundarycondition has to be satisfied almost everywhere on T. Since the target circles are uniformly bounded it is clearthat solutions in Hp automatically belong to H∞.

Theorem 5.2 Let r and c be Holder continuous on T and assume that the Riemann-Hilbert problem |w−c| =r is regular. Then the Nevanlinna-parametrization gives rise to a homeomorphism between

(a) the set of finite Blaschke products (endowed with the topology of uniform convergence) and the solutions win the disk algebra H∞ ∩ C

(b) the set of inner functions (endowed with the topology of H∞) and the solutions w in the Hardy space H∞.

P r o o f. Both assertions follow from the fact that the coefficients of the Nevanlinna parametrization are(Holder) continuous and that for every t ∈ T the fiber map Nt is a homeomorphism between the target cir-cle Mt and the unit circle.

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In particular the result shows that regular problems also have “irregular” solutions. This is not so in the singularcase, where we have a unique bounded solution which is automatically continuous (see Corollary 6.2 below).

For the sake of completeness we close this section with (more or less quoting) a result on the regularity ofsolutions.

Theorem 5.3 Let r and c belong to Cα(T) with 0 < α < 1 and let w be a solution of |w − c| = r in the discalgebra H∞ ∩ C. Then the following assertions hold:

(i) If r, c ∈ Ck+α(T) with k ∈ Z and k ≥ 0, then w belongs to H∞ ∩ Ck+α.

(ii) If r, c ∈W kp (T) with k ∈ Z, k ≥ 1, and 1 < p <∞, then w belongs to H∞ ∩W k

p .

P r o o f. For k = 0 the first statement (i) has already been shown. The case k ≥ 1 and the second assertionfollow from regularity results for solutions of general nonlinear Riemann-Hilbert problems with continuouslydifferentiable boundary condition, see [27], Section 2.8.

Alternatively, one can start from the regularity of the best H∞ approximation (see [17], Chap. 7.2), whichyields the regularity of one solution to the Riemann-Hilbert problem. In the singular case this is the uniquesolution and we are done. In the regular case we use mapping properties of the Hilbert transform to get regularityof the solutions to the related linear problems, which in turn yields regularity of the coefficients in the Nevanlinnaparametrization, and finally regularity of all solutions.

6 Relations to the Nevanlinna-Pick Problem

In this section we shortly discuss the interplay between Riemann-Hilbert problems and the interpolation prob-lem of Nevanlinna and Pick. The classical problem consists in finding all unimodularly bounded holomorphicfunctions w in D which satisfy the interpolation conditions

w(zj) = wj , j = 1, . . . , n, (14)

where the interpolation nodes zj and values wj are given in D. For multiple points zj the conditions have to bemodified in the sense of Hermite interpolation. If, in particular, z1 = . . . = zn = 0 we arrive at the Cartheodory-Fejer problem of finding unimodularly bounded holomorphic functions with a prescribed initial part of the Taylorseries.

In the context of Riemann-Hilbert problems |w − c| = r it is natural to pose the interpolation problems forfunctions in the set

A := {w ∈ H∞ : |w − c| ≤ r a.e. on T}.

Since these boundary restrictions are more flexible it is easy to remove the interpolation conditions and thepeculiarities of the Hermite and Caratheodory-Fejer cases disappear.

We start with a result which relates the number #A of elements in A to the classification of the correspondingRiemann-Hilbert problem.

Lemma 6.1 Let A be defined by (10) with c, r ∈ Cα(T). Then the Riemann-Hilbert problem |w − c| = r isregular, singular, or void if (and only if) #A > 1, #A = 1, or #A = 0, respectively.

P r o o f. If #A = 0 the Riemann-Hilbert problem must obviously be void. If #A > 0 Lemma 4.6 guaranteesthat the Riemann-Hilbert problem has a solution w in H∞ ∩ Cα, so it must be regular or singular. Clearlyregularity implies that #A > 1, and it remains to prove the converse, i.e. #A = 1 for all singular problems.

Let w0 ∈ H∞ ∩ Cα be a solution of the singular problem |w − c| = r (which exists by Theorem 4.2) andassume that w ∈ A is a function different from w0. Denote by f the inner normal at Mt at w0(t). Then thefunction w − w0 is in H∞ and satisfies Im

(f (w − w0)

)≥ 0 on T. Since wind f = windMw0 < 0 we infer

from Lemma 3.4 (ii) that w − w0 ≡ 0.

Corollary 6.2 If the set A is not empty, then it contains a Holder continuous function (namely the solution ofthe corresponding Riemann-Hilbert problem). Thus, in contrast to regular problems, any (generalized) solutionw ∈ H∞ is automatically Holder continuous.

14 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

In order to employ Lemma 6.1 for Nevanlinna-Pick interpolation problems in the set A we choose an interpo-lation polynomial p which satisfies (14), set q(z) :=

∏nj=1(z − zj), and define

M := (M − p)/q, w := (w − p)/q, w := p+ q w. (15)

Then w belongs to A and satisfies the interpolation conditions if and only if w belongs to

A := {w ∈ H∞ : |w − (c− p)/q| ≤ r/|q| a.e. on T}.

Analogously w is a solution of the Riemann-Hilbert problem trw ⊂M and satisfies (14) if and only if w solvestrw ⊂ M . In this case the winding numbers of w and w are related by windM w = windfM w + n.

Lemma 6.3 Let c, r ∈ Cα(T) with 0 < α < 1 and r > 0, and let the polynomials p and q be chosen asabove.

(i) Let the Riemann-Hilbert problem with target manifold (M−p)/q be regular. Then the interpolation problem(14) has infinitely many solutions in A, and there exist solutions w ∈ H∞ ∩ Cα which satisfy |w − c| = ron T and windM w = n. There are no such solutions with windM w < n.

(ii) Let the Riemann-Hilbert problem with target manifold (M − p)/q be singular. Then the interpolationproblem is uniquely solvable. The solution belongs to H∞ ∩ Cα, it satisfies |w − c| = r on T, andwindM w < n.

(iii) If the Riemann-Hilbert problem with target manifold (M − p)/q is void the interpolation problem has nosolution.

P r o o f. The result follows immediately from Theorem 1.2, Theorem 4.2, and Lemma 6.1, using the transfor-mations (15).

Corollary 6.4 If the interpolation problem (14) has a solution with |w− c| < r, then it also admits a solutionwith |w − c| = r on T and windM w = n.

The Nevanlinna parametrization N for solutions of Nevanlinna-Pick or Caratheodory-Fejer interpolationproblems can also easily be obtained from the parametrization N of the associated Riemann-Hilbert problemtr w ⊂ M . Namely, let

NB =PB +Q

RB + S, NB =

PB + Q

RB + S.

Using the transformation (15) we get the relations

P = pR+ qP , Q = pS + qQ, R = R, S = S.

Since PS − QR = q (P S − QR) assertion (iii) of Theorem 5.1 tells us that the function PS − QR in theNevanlinna parametrization for interpolation problems vanishes in D exactly at the nodes zj .

In Section 5 we derived the Nevanlinna parametrization of solutions to the Riemann-Hilbert problem startingfrom three solutions with winding number zero about M . We now show how the parametrization can be obtainedfrom one solution w1 with arbitrary winding number n ≥ 0.

Let p be a polynomial such that w1 − p has a zero of order n at z = 0. We consider the Riemann-Hilbertproblem with the target manifold M := t−n(M − p). The function w1 defined by w1(z) := z−n(w1(z)− p(z))is a solution of this problem and has winding number zero about M . Using the reduction to a linear problemin Section 4 we get three solutions of trw ⊂ M with winding number zero, which allows to construct theNevanlinna parametrization

N B =P B + Q

R B + S

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of the transformed problem. Using the reverse transformation w(z) = zn w(z) + p(z) we obtain

w(z) = p(z) + zn P (z)B(z) + Q(z)

R(z)B(z) + S(z). (16)

This is not the Nevanlinna parametrization of the original problem, since it only yields solutions with windingnumber n for which w − p has a zero of order not less than n at z = 0. However, we can use it to find threesolutions of the original problem with winding number zero, to which then the construction of Section 5 isapplicable.

Since the function −R/S is holomorphic in D and has modulus strictly less than one in D, the Caratheodory-Fejer problem

B∗(z) = − R(z)

S(z)+O(zn) (17)

admits solutions B∗ which are Blaschke products of exact degree n (see Corollary 6.4). We fix one such solutionand define

w0(z) = p(z) + zn P (z) + Q(z)B∗(z)

R(z) + S(z)B∗(z), z ∈ D. (18)

The denominator R + SB∗ has a zero of order not less than n at z = 0. Since |R| < |S| and |B∗| = 1 on T,Rouches theorem guarantees that R + SB∗ has the same number of zeros as SB∗, and, because S has no zeros,this number is n. Consequently z = 0 is the only zero of R + SB∗ in D and has exact order n. Thus w0 isholomorphic in D and continuous on D.

To prove that w0 is a solution of the original Riemann-Hilbert problem we remark that

w0(t) = p(t) + tnP (t)b+ Q(t)

R(t)b+ S(t)∈Mt,

for all b ∈ T, and with b := 1/B∗(t) the assertion follows.It remains to show that the solution w0 has winding number zero about M . For this end we associate with

two solutions w0 and w1 the family of uniquely defined straight lines Lt, t ∈ T such that Lt contains the pointsw0(t), w1(t) and is tangent toMt in case that both points coincide. The family Lt depends continuously on t ∈ Tand its winding number is said to be the generalized winding number wind (w1−w0) of the difference w1−w0.Note that the family of lines Lt is not necessarily orientable, such that wind (w1 − w0) ∈ 1

2 Z.A little geometric imagination (or a homotopy argument) shows that for two solutions with winding numbers

m and n about M always wind (w1 − w0) = (m+ n)/2.We now consider the difference of the given solution w1 with winding number n and the solution w0 con-

structed above. Since, in representation (16), w1 corresponds to a Blaschke product B ≡ c of order zero, weget

w1(z)− w0(z) = zn

[c P (z) + Q(z)

c R(z) + S(z)− P (z) + Q(z)B∗(z)

R(z) + S(z)B∗(z)

]

=zn

R(z) + S(z)B∗(z)· 1

c R(z) + S(z)·[Q(z)R(z)− P (z)S(z)

]·[1− cB∗(z)

].

It has already been shown that the first factor does not vanish on D. By the properties of the Nevanlinnaparametrization (see assertion (iii) of Theorem 5.1) this also holds for the second and the third factor. The lastfactor has no zeros in D and exactly n zeros on T. Consequently the generalized winding number wind (w1−w0)satisfies

n

2= wind (w1 − w0) =

12

(windM w1 + windM w0) =n

2+

12

windM w0,

16 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

and hence windM w0 is zero.In summary three solutions of the Caratheodory-Fejer problem (17) of degree n yield three solutions of the

Riemann-Hilbert problem with winding number zero, from which the full Nevanlinna parametrization can beobtained.

7 Classification of Problems

In this section we collect several criteria for deciding to which class a given problem |w − c| = r belongs. Thefollowing two theorems are applicable if at least one holomorphic function with |w − c| ≤ r is known.

Theorem 7.1 Let r, c ∈ Cα(T), 0 < α < 1, with r > 0. Then the following statements are equivalent.

(i) The Riemann-Hilbert problem |w − c| = r is regular.

(ii) There exists a continuous holomorphic function w ∈ H∞∩C such that |w−c| = r on T and windMw ≥ 0.

(iii) There exist two different bounded holomorphic functions w1, w2 ∈ H∞ such that |wj − c| ≤ r almosteverywhere on T.

(iv) There exists a bounded holomorphic function w ∈ H∞ such that |w − c| ≤ r almost everywhere on T and|w − c| < r on a subset of positive measure.

P r o o f. By Lemma 6.1 any of the four conditions guarantees that the problem is not void. Then (i) ⇔ (ii)follows from Theorem 1.2 and Theorem 4.2, the equivalence (i) ⇔ (iii) results from Lemma 6.1, to prove theimplication (iii)⇒ (iv) we remark that w := (w1 +w2)/2 satisfies |w(t)− c(t)| < r(t) whenever w1(t) 6= w2(t)(which holds almost everywhere on T because w1 6≡ w2), and (iv) ⇒ (i) is a consequence of Theorem 4.2 andLemma 6.1.

Theorem 7.2 Let r, c ∈ Cα(T), 0 < α < 1, with r > 0. Then the following statements are equivalent.

(i) The Riemann-Hilbert problem |w − c| = r is singular.

(ii) There exists a continuous holomorphic function w ∈ H∞∩C such that |w−c| = r on T and windMw < 0.

(iii) There exists a bounded holomorphic function w ∈ H∞ such that |w − c| ≤ r almost everywhere on T, butno such function with |w − c| < r on a subset of positive measure.

P r o o f. Again the problem cannot be void. The equivalence (i) ⇔ (ii) follows from Theorems 1.2 and 4.2,and (i) ⇔ (iii) is a consequence of Theorem 7.1 (iv).

If no additional information is available one can apply the “Hankel test”. It rests on Nehari’s theorem, sayingthat theH∞ approximation error for the function c is equal to the Norm of the Hankel operatorH(c) with symbolc (see [16], Theorem 1.6.8, for instance). In one (of several common) definitions this operator maps H2(T) toits orthogonal complement L2(T)H2(T) by f 7→ (I − P )cf , where P is the orthogonal projection of L2(T)onto H2(T).

Since any problem |w−c| = r is linearly equivalent to a problem with r ≡ 1 (via explicit formulas), it sufficesto consider the latter case.

Theorem 7.3 Let c ∈ Cα(T). Then the Riemann-Hilbert problem |w − c| = 1 is regular, singular orvoid, depending on whether the Hankel operator H(c) with symbol c satisfies ‖H(c)‖ < 1, ‖H(c)‖ = 1 or‖H(c)‖ > 1, respectively.

P r o o f. The statement follows from Theorem 7.1 (iv) and Theorem 7.2 (iii) by Nehari’s theorem.

For practical purposes the following criterion is useful, which, in principle, allows to classify any given prob-lem which is not singular.

We assume that Tnc is an arbitrary sequence of trigonometric polynomials of degree not exceeding n con-verging uniformly to c. If c ∈ Cα(T) we can take the finite sections of its Fourier series, for instance. With the

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function Tnc(t) =∑n

k=−n cktk we associate the Hankel matrix (the coefficients ck may also depend on n which

we neglect in the notation)

Hn(c) :=

c−1 c−2 c−3 . . . . . . c−n

c−2 c−3 . . . . . . c−n 0c−3 . . . . . . . . . 0 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .c−n 0 0 . . . 0 0

.Then the norm of the Hankel operator H(Tnc) is equal to the spectral norm of Hn(c) and can be computednumerically as the largest singular value of this matrix.

Theorem 7.4 Let c ∈ Cα(T) and assume that the Hankel matrices Hn(c) and the trigonometric polynomialsTnc are defined as above.

(i) The Riemann-Hilbert Problem |w − c| = 1 is regular if there exists a positive integer n such that

‖Hn(c)‖ < 1− ‖c− Tnc‖∞, (19)

and it is void if there exists a positive integer n such that

‖Hn(c)‖ > 1 + ‖c− Tnc‖∞. (20)

(ii) If the problem |w − c| = 1 is not singular then one of the conditions (19) or (20) holds for all sufficientlylarge n.

P r o o f. Since ‖H(c)‖ ≤ ‖c‖∞ we can estimate∣∣‖H(c)‖−‖Hn(c)‖

∣∣ ≤ ‖c−Tnc‖∞, so that both assertionseasily follow from Theorem 7.3.

Another more direct algorithm for classifying problems |w − c| = r in the case where c and r2 are rationalfunctions will be given in the second part of the paper.

8 Sharpness of the Results

At the end we give two counterexamples which illustrate that the main results do in general not hold for continu-ous functions c and r.

Lemma 8.1 There exist continuous functions r and c with r > 0 such that the Riemann-Hilbert problem|w − c| = r admits no solution in H∞ ∩ C.

P r o o f. We consider a conformal map f of the unit disk D onto the domain

G := {z = x+ i y : 0 < x < 1/(1 + y2)}.

The real part of f is continuous on D, while the imaginary part is unbounded. We set r := exp(Re f) and showthat the Riemann-Hilbert problem |w| = r has no continuous solution.

Assume first that w ∈ H∞ ∩ C were a solution of |w| = r. We can assume that w has no zeros, sinceotherwise a finite Blaschke product formed by the zeros of w can be factored off, which does not change theboundary condition. Then logw (with any continuous branch of the logarithm) is well-defined and belongs toH∞ ∩ C. Further Rew = log r = Re f on T, which implies that Imw − Im f = const, a contradiction.

Even if a problem with continuous c and r has “enough” continuous solutions their traces need not cover thetarget manifold.

Lemma 8.2 There exist continuous functions r and c with r > 0 such that the traces of continuous solutionof the Riemann-Hilbert problem |w − c| = r are dense on its target manifold M , but do not cover M .

18 Ch. Glader and E. Wegert: Circular Riemann-Hilbert Problems

P r o o f. Let f be the conformal map of the unit disk D onto the domain (see Figure 2)

G :={z = 2− r eiϕ : −π < ϕ ≤ π, 0 < r <

√π2/(π2 − ϕ2)

}normalized so that f(0) = 2 and f ′(0) > 0. Then, f extends continuously onto T \ {1}, Re f ≥ 1 on D,and f(z) → ∞ as z → 1. Consequently, the functions λ/f with λ ∈ R are (the) continuous solutions of ahomogeneous linear Riemann-Hilbert problem with continuous coefficients and index zero.

Figure 2

The corresponding target curves are straight lines which do not meet the point i , and the map z 7→ 1/(z − i)sends them to a family Mt of (target) circles which pass the origin, the point i, and the variable point f(t)/

(1−

i f(t)). The center c(t) and the radius r(t) of Mt depend continuously on t.

We set w0 := i = const, w∞ := 0 = const, wλ := f/(λ − i f) for λ ∈ R \ {0}. All functions wλ arecontinuous on D, satisfy the boundary condition |w − c| = r and have winding number zero about M . Usingthe same reasoning as in the uniqueness part of the proof of Theorem 1.2 (page 10) we see that the family wλ,λ ∈ R, contains all continuous solutions of |w − c| = r with winding number zero about M . Since w0(1) = iand wλ(1) = 0 for all other values of λ, not all of M is covered by the traces of these solutions. More precisely,the number of traces passing a point (t, z) ∈ M is one if t 6= 1, or t = 1 and z = i; it is zero if t = 1 andz /∈ {0, i}; and it is infinite if t = 1 and z = 0.

Acknowledgement. We would like to thank the referee for several suggestions to improve the presentation.

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