Bergman orthogonal polynomials and the Grunsky matrix Bernhard Beckermann, Nikos Stylianopoulos

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Bernhard Beckermann, Nikos Stylianopoulos. Bergman orthogonal polynomials and the Grunsky matrix. 2016. ￿hal-01325026v1￿

HAL Id: hal-01325026 https://hal.archives-ouvertes.fr/hal-01325026v1 Preprint submitted on 1 Jun 2016 (v1), last revised 5 Jul 2016 (v2)

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by

Bernhard Beckermann1 and Nikos Stylianopoulos2

Abstract

By exploiting a link between Bergman orthogonal polynomials and the Grunsky ma- trix, probably first observed by K¨uhnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G, and the associated conformal maps. For quasiconformal boundaries, this approach allows for new insights for Bergman polynomials.

Key words: Bergman orthogonal polynomials, Faber polynomials, Conformal mapping, Grun- sky matrix, Bergman shift, Quasiconformal mapping. Subject Classifications: AMS(MOS): 30C10, 30C62, 41A10, 65E05, 30E10.

1 Introduction and statement of the main results

Let G be a bounded simply connected domain in the complex plane C, with Jordan boundary Γ. We define, for n 0, the so-called Bergman orthogonal polynomials ≥ n pn(z)= λnz + terms of smaller degree, λn > 0, satisfying

p ,p 2 := p (z)p (z) dA(z)= δ , h n miL (G) n m m,n ZG where dA(z) = dxdy denotes planar Lebesgue measure. There is a well developed theory re- garding Bergman polynomials. For their basic properties, and the asymptotic behavior including that of their zeros, see for instance [3, 28, 7, 20, 8, 9, 5, 26, 21, 27] and the references therein. In describing these we need two conformal maps. Denote by D the open unit disk, by D∗ the exterior of the closed unit disk, by Ω the exterior of the closure of G, and consider the Riemann map from Ω onto D∗ −1 w = φ(z)= φ1z + φ0 + φ−1z + ..., with inverse map −1 −1 z = ψ(w)= φ (w)= ψ1w + ψ0 + ψ−1w + ..., where γ := φ = 1/ψ = 1/ψ′( ) > 0 is the inverse of the logarithmic capacity of G. We make 1 1 ∞ also use of the n-th Faber polynomial Fn. This is the polynomial part of the Laurent expansion 1Laboratoire Painlev´eUMR 8524 (AN-EDP), UFR Math´ematiques – M3, Univ. Lille, F-59655 Villeneuve d’Ascq CEDEX, France. E-mail: [email protected]. Supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). 2Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus. E-mail: [email protected]. Supported in part by the University of Cyprus grant 3/311-21027.

1 at of φ(z)n, and hence F is of degree n, with leading coefficient γn. The corresponding ∞ n Grunsky coefficients bℓ,n = bn,ℓ are defined by the generating series

∞ ∞ ∞ ψ(w) ψ(v) 1 log − = b w−ℓv−n = F (ψ(w)) wn v−n, (1.1) ψ′( )(w v) − n,ℓ − n n − n=1 n=1  ∞ −  X Xℓ=1 X   which is analytic and absolutely convergent for w > 1, v > 1, see, e.g., [18, Chapter 3, Eqns | | | | (8) and (10)].

In what follows it will be convenient to work with the normalized Grunsky coefficients

Cn,k = Ck,n = √n + 1 √k + 1 bn+1,k+1, for n, k = 0, 1, 2, ..., (1.2) for which the Grunsky inequality [18, Theorem 3.1] reads as follows: For any integer m 0 and ≥ any complex numbers y0,y1, ..., ym there holds

∞ m m 2 C y y 2. (1.3) n,k k ≤ | k| n=0 k=0 k=0 X X X

We note that (1.3) holds under the sole assumption that G is a continuum, i.e., closed and connected, and Ω is the component of C G that contains . \ ∞ Under various assumptions on Γ many results are scattered over the literature, saying that the Bergman polynomial pn behaves like a suitably normalized derivative of the Faber polynomial Fn+1, namely like

′ Fn+1(z) n + 1 n+1 n fn(z) := = γ z + smaller degree. (1.4) √π√n + 1 r π Taking derivatives with respect to w in (1.1), using (1.2) and comparing like powers of v leads, for w > 1, to the formula | | ∞ ′ n + 1 n ℓ + 1 −(ℓ+2) rn(w) := ψ (w)fn(ψ(w)) w = w Cℓ,n. (1.5) − r π − r π Xℓ=0 The Grunsky inequality (1.3) allows us to conclude3 that r L2(D∗), and more precisely by n ∈ choosing y0 = y1 = ... = yn−1 = 0, yn = 1 we obtain

∞ 2 2 ε := r 2 D∗ = C 1. (1.6) n k nkL ( ) | j,n| ≤ Xj=0

We note that, though there is a different normalization for fn, the quantity εn in (1.6) coincides 2 with that of [26, Eqn. (2.21)]. Also, it is interesting to observe that ε = 1 f 2 , see n − k nkL (G) Lemma 2.2 below.

Most of the results of this paper require the additional assumption that the boundary Γ is quasiconformal. We refer the reader to, e.g., [18, 9.4] for a definition and elementary properties, § and recall that a piecewise analytic Jordan curve Γ is quasiconformal if and only if it does not have cusps. Also, a quasiconformal curve is a Jordan curve, with two-dimensional Lebesgue

3We do not need further assumptions on the boundary like Γ being rectifiable, compare with [26, Lemma 2.1].

2 measure zero [1, Section B.2.1]. In our considerations in 2 the following fact will be important § [18, Theorem 9.12 and Theorem 9.13]: we can improve the Grunsky inequality (1.3) exactly for the class of quasiconformal curves, by inserting a factor strictly less than one on the right-hand side of (1.3).

For quasiconformal Γ, the following inequalities have been established by one of the present authors in [26, Lemma 2.4 and Theorem 2.1]:

n + 1 γ2n+2 εn 1 2 = (εn). (1.7) ≤ − π λn O In other words, the leading coefficient λ of p behaves like that of f if and only if ε 0. By n n n n → using (1.7) and the Pythagoras theorem, is is readily seen that

2 f p 2 = (ε ), (1.8) k n − nkL (G) O n that is, for ε 0 we also get L2 asymptotics on the support of orthogonality for p . Further- n → n more, by employing the argument of [22, p. 2449], the relation (1.8) will lead to the relative asymptotics fn(z) =1+ (√εn), (1.9) pn(z) O at least for z outside the closed convex hull conv(G) of G. Note that, by Fejer’s theorem [7, Theorem 2.1], pn is zero free outside conv(G). For the sake of completeness, we will provide an alternate proof of (1.7) and (1.8) in Remark 2.5 below, based on our Theorem 2.4. Moreover, a sharper result related to (1.9) will be established in Theorem 1.1 below.

Here we give a short review of some cases where the behavior of εn is known. Further details are given in 3 below. For analytic Γ, the function ψ has an analytic and univalent continuation § to w > ρ for some ρ [0, 1). We then write Γ U(ρ) for the smallest such ρ. The asymptotic | | ∈ ∈ behavior of Bergman polynomials for Γ U(ρ) has been first derived by Carleman in [3] where, ∈ in particular, he showed that

2n+2 n + 1 γ 2n 0 < 1 2 = (ρ ). (1.10) − π λn O

For Γ U(ρ), K¨uhnau [15, Eqn. (9)] established the inequality C ρℓ+n+2 and hence, ∈ | ℓ,n| ≤ from (1.6), ε = (ρ2n). Thus (1.10) can be also deduced from (1.7) by means of the Grunsky n O coefficients. The particular case of an ellipse discussed in 2.1 below shows that (1.10) is not 4n+4 § sharp for ellipses, because in this case εn = ρ . For a recent discussion of asymptotics inside G when Γ U(ρ) we refer to Mi˜na-D´ıaz [16]. ∈ Suetin considered Γ in the class (p, α), meaning that there is a parametrization of Γ C having a derivative of order p 1, which is H¨older continuous with index α (0, 1). Under ≥ ∈ the assumption Γ (p + 1, α), p 0, he showed in [28, Lemma 1.5] that ε = (1/nβ) with ∈ C ≥ n O β = 2p + 2α. In cases where β > 1, the upper bound of (1.7) can be found in [28, Theorem 1.1].

Finally, the case of piecewise analytic Γ without cusps has been considered recently by the second author, who showed in [26, Theorem 2.4] that ε = (1/n). An example of two n O overlapping disks considered by Mi˜na-D´ıaz [17] shows that this estimate cannot be improved, since lim infn nεn > 0, for this particular Γ.

3 For easy reference below we summarize (ρ2n), if Γ U(ρ), O 2(p+α) ∈ εn = (1/n ), if Γ (p + 1, α), (1.11)  O ∈C (1/n), if Γ piecewise analytic without cusps.  O In order to include all three preceding cases, we will assume below that ε = (1/nβ), for some β > 0. (1.12) n O

Using the reproducing kernel property in L2(D∗) we find for w > 1 that | | 1 rn(w)= rn(u)K(u, w)dA(u), where K(u, w)= . D∗ π(wu 1)2 Z − 2 By the Cauchy-Schwarz inequality, (1.6), and the fact that K(w, w) = K( , w) 2 ∗ , we k · kL (D ) conclude that εn/π r (w) r 2 D∗ K( , w) 2 D∗ = . (1.13) | n | ≤ k kL ( ) k · kL ( ) w 2 1 |p| − Therefore, it follows from (1.5), for w = φ(z), that π f (z) π r (w) n 1= n (1.14) n + 1 φ′(z)φ(z)n − n + 1 wn r r tends to zero uniformly on compact subsets of Ω with a geometric rate.

Our first main result describes strong asymptotics for Bergman polynomials outside the support of orthogonality. Theorem 1.1. Let Γ be quasiconformal, and assume that ε = (1/nβ), for some β > 0. Then n O π p (z) √ε n 1= ( n ), n , (1.15) n + 1 φ′(z)φ(z)n − O nβ/2 →∞ r uniformly on compact subsets of Ω.

This result should be compared with [26, Theorem 1.2] for piecewise analytic Γ without cusps (β = 1), or [28, Theorem 1.4] for sufficiently differentiable Γ where, in both cases, the rate is given by (√ε ), the same rate as in (1.9). That is, in these two cases, (1.15) yields O n an improvement of order 1/nβ/2 of the estimated rate of convergence. When Γ U(ρ), we ∈ may also apply Theorem 1.1 for every β > 0, while Carleman [7, Theorem 2.2] gives the rate √nε = (√nρn), see also [5, page 1983] and [16, Theorem 1]. We should emphasize, however, n O that our estimate (1.15) is only valid on compact subsets of Ω.

In contrast to [28] and related work for (complex) Jacobi matrices (see for instance [24] or [2, 3.5] and the references therein), our method of proof is not based on operator theory techniques, § like determinants of the identity plus a trace class perturbation, and thus here (besides (1.12) with β > 0) we do not need any further assumption on the decay of ε . { n} Our second main result concerns the multiplication operator H, sometimes called Bergman shift, defined by

n+1 zp (z)= p (z)H , with H = zp ,p 2 . (1.16) n j j,n j,n h n jiL (G) Xj=0 4 Notice that, by orthogonality, Hj,n = 0 for j >n + 1, and thus the infinite matrix H has upper Hessenberg form. From the orthogonality of the Bergman polynomials one deduces that H represents a bounded linear operator in ℓ2, with H sup z : z G , cf. [4, 29]. k k≤ {| | ∈ } § Theorem 1.2. Let Γ be quasiconformal. Then there exists a constant c(Γ) such that, for all k 1 and n 0, ≥− ≥ n + 1 H ψ c(Γ) (k + 2) max ε , ..., ε , ε . n−k,n − n k + 1 −k ≤ n−k−1 n n+1 r −  

If we assume further in Theorem 1.2 that (1.12) holds, then it follows that, for any k 1, ≥− n + 1 1 H ψ = ( ), n . (1.17) n−k,n − n k + 1 −k O nβ →∞ r −

This result should be compared with the asymptotic results [21, Theorem 2.1, Theorem 2.2], where the same rate is given for k = 1 and n , but for fixed k 0 only the rate − → ∞ ≥ (1/nβ/2) is obtained. Similarly, for Γ U(ρ), Theorem 1.2, in view of (1.11), yields the O n→∞ ∈ rate (ρ2n) for any fixed k, improving thus by ρn the rate established in [21, Theorem 2.3] in O the case k 0. ≥ The reminder of the paper is organized as follows. In 2 we suggest a new matrix formalism § in order to state and prove (in Lemma 2.2 and Corollary 2.3) K¨uhnau’s link between the Gram matrix ( f ,f 2 ) and the Grunsky matrix. We then provide, in Theorem 2.4, an estimate h j kiL (G) j,k in terms of ε for the coefficients in the change of basis from p ,p , ..., p to f ,f , ..., f , n { 0 1 n} { 0 1 n} and vice-versa. This allows us to present, in Remark 2.5, alternate proofs of the relations in (1.7) and (1.8). We conclude 2 with a detailed discussion of an ellipse, where several of the § above asymptotic results are shown to be sharp.

In 3 we gather together known results from the literature relating spectral and asymptotic § properties of the Grunsky operator and the size of εn to the smoothness of Γ. In particular, we conclude that the Grunsky operator for domains with corners is not compact.

Finally, in 4, we give the proofs of our main Theorems 1.1 and 1.2. §

2 The Grunsky matrix and quasiconformal boundaries

The aim of this section is to describe the size of the Fourier coefficients R C of f in the j,n ∈ n orthonormal basis p , that is, { n} n f (z)= p (z)R ,R = f ,p 2 . (2.1) n j j,n j,n h n jiL (G) Xj=0

It will be convenient to use matrix calculus. In what follows we denote by e , j 0, the j ≥ j-th canonical vector in Cn as well as in ℓ2, the size depending on the context. The action of a bounded linear operator B : ℓ2 ℓ2 can be described through matrix products, where →

5 we identify B with its infinite matrix ( Be , e 2 ) . We will also consider the operator h k jiℓ j,k=0,1,... Π : Cn ℓ2 with matrix representation n → I Π = C∞×n, with adjoint Π∗ = I 0 Cn×∞, n 0 ∈ n ∈     so that ∗ Bn = ΠnBΠn = (Bj,k)j,k=0,1,...,n−1, gives the principal submatrix of order n of B (sometimes called the n-th finite section). Note that B = supn Bn = limn→∞ Bn , where denotes the euclidian vector norm as well k k k k k k k · k 2 as the subordinate matrix (operator) norm, i.e., Bn is the largest of the eigenvalues of the ∗ k k Hermitian matrix BnBn.

Then, it follows from the Grunsky inequality (1.3) that the infinite Grunsky matrix C = (C ) satisfies C CΠ 1 for all n, and thus C 1. We will also use n,k n,k=0,1,... k nk ≤ k nk ≤ k k ≤ frequently the identity ε = Ce 2, (2.2) n k nk contained already implicitly in (1.6). Moreover, according to [18, Theorems 9.12 and 9.13], Γ is quasiconformal if and only if C < 1 (and more precisely Γ is κ-quasiconformal if and only if k k C κ). k k≤ We recall now from [15, 5] a geometry, where C can be computed explicitly. Further § k k geometries are discussed in [15, 5 and 6]. § § Example 2.1. If Γ is piecewise analytic with a corner with outer angle ωπ, 0 <ω< 2, then K¨uhnau showed, with the help of the Golusin inequality [18, 3.2], that C 1 ω , with § k k ≥ | − | equality C = 1 ω = 2/m for all regular polygons with m 3 vertices.  k k | − | ≥

In what follows we set Rj,n = 0 for j>n, so that the infinite matrix R is upper triangular. As a consequence, any principal submatrix of order k n of (R )−1 is given by (R )−1. We may ≤ n k therefore write for 0 j k, without ambiguity, R−1 = (R−1) = ((R )−1) for the entries ≤ ≤ j,k j,k k j,k of the (possibly unbounded) upper triangular infinite matrix R−1, and get from (2.1) that

n −1 pn(z)= fj(z)Rj,n. (2.3) Xj=0

We are now ready to make the link between the Grunsky matrix, Faber polynomials and the Bergman polynomials. Such a link has been probably first studied by K¨uhnau in [14], though the identity (2.4) below can be found in the work of Johnston [11, Lemma 4.3.8] and Suetin [28, p. 13], without alluding, however, to the relation with the Grunsky matrix. The same identity is mentioned without proof in [23, Eqn. (2.9)].

Lemma 2.2. Assume that Γ is of two-dimensional Lebesgue measure zero. Then, for all n, k ≥ 0, there holds

f ,f 2 = δ r , r 2 D∗ (2.4) h n kiL (G) n,k − h n kiL ( ) = δ e∗C∗Ce . (2.5) n,k − k n

6 Proof. For r > 1 we consider the level set G := C ψ(w) : w r , which has analytic r \ { | | ≥ } boundary ∂Gr. A proof of Lemma 2.2 is based on the use of Green’s formula 1 f(u)g′(u)dA(u)= f(u)g(u)du, 2i ZGr Z∂Gr where f, g are analytic on Gr. For the sake of completeness we reproduce here the idea of the proof given in [14]. Comparing like powers of v in (1.1) and using (1.2), we have for w > 1 that | | k+1 ∞ −(ℓ+1) Fk+1(ψ(w)) w w = + Cℓ,k. (2.6) π(k + 1) π(k + 1) π(ℓ + 1) Xℓ=0 Notice that taking derivativesp in (2.6) leadsp to (1.5). p

We thus obtain by applying Green’s formula

′ 1 Fn+1(z) Fk+1(z) fn(z)fk(z)dA(z)= dz r 2i r π(n + 1) π(k + 1) ZG Z∂G 1 ψ′(w)F ′ (ψ(w)) F (ψ(w)) = n+1 kp+1 dw.p 2i π(n + 1) π(k + 1) Z|w|=r Inserting (1.5) and (2.6) in the latterp integral we getp the expression

∞ ∞ 1 n + 1 ℓ + 1 wk+1 w−(j+1) wn w−(ℓ+2)C + C dw. 2i π − π ℓ,n j,k |w|=r r r π(k + 1) j=0 π(j + 1) Z  Xℓ=0  X  Here the sums are uniformly convergent in w = r andp we can thus integratep term by term. | | Since 1 r2ℓ+2 wjwℓ+1dw = wjw−ℓ−1dw = r2ℓ+2 δ , 2πi 2πi j,ℓ Z|w|=r Z|w|=r for j, ℓ Z, we therefore arrive at ∈ ∞ 2k+2 −2ℓ−2 f ,f 2 = δ r C C r . (2.7) h n kiL (Gr) n,k − ℓ,k ℓ,n Xℓ=0

Clearly, the polynomials f and f are continuous in C and the closure G Γ of G is the n k ∪ intersection of the monotone family Gr, for r> 1. Hence, we conclude that

2 2 2 lim fn,fk L (Gr) = fn,fk L (G∪Γ) = fn,fk L (G), r→1+h i h i h i the last equality follows from our assumption on Γ.

The relation C 1 implies that Ce = (C ) ℓ2 for all k 0 and hence (C C ) k k≤ k j,k j ∈ ≥ j,k j,n j ∈ ℓ1, implying that the term on the right-hand side of (2.7) has a (finite) limit for r 1+, given → by δ e∗C∗Ce . Moreover, by integrating (1.5) over rD∗ and D∗ we have n,k − k n ∞ ∞ −2ℓ−2 lim rn, rk L2(rD∗) = lim Cℓ,kCℓ,nr = Cℓ,kCℓ,n = rn, rk L2(D∗), r→1+h i r→1+ h i Xℓ=0 Xℓ=0 which, in view of (2.7), implies both (2.4) and (2.5). 

7 It should be noted that in the proof we did not use any assumptions on the boundary Γ of G, other that it should have zero Lebesgue measure.

Next we will apply Lemma 2.2 to quasiconformal Γ. As mentioned before, Γ then has two-dimensional Lebesgue measure zero and satisfy C < 1 [18, Theorem 9.13]. Using (2.1) k k and the orthonormality of Bergman polynomials, we find that

min(n,k) f ,f 2 = R R . h n kiL (G) j,k j,n Xj=0 Recalling that B = B∗ = B∗B , for any operator B acting on ℓ2, the following result is k k k k k k an easy consequence of Lemma 2.2. p Corollary 2.3. Let Γ be quasiconformal. Then the infinite Gram matrix M = ( f ,f 2 ) h n kiL (G) k,n=0,1,... can be written as M = R∗R = I C∗C. − Furthermore, M and R represent two bounded linear operators acting on ℓ2 with norms 1, ≤ while C is such that C < 1. In addition, both M and R are boundedly invertible, with k k M −1 = R−1 2 1/(1 C 2). k k k k ≤ − k k

It is interesting to note that, according to the upper triangular structure of R, we also have for all n 0 that M = R∗ R , the Cholesky decomposition of the Hermitian positive definite ≥ n n n Mn.

−1 Our next result provides various estimates for the coefficients Rj,k and Rj,k in (2.1) and (2.3).

Theorem 2.4. Let Γ be quasiconformal. Then we have, for all j 0, ≥ 2 ∗ ∗ ∗ −1 ∗ −1 ∗ C max ej (I R ) , ej (R I) ej (R R ) εj k k . (2.8) k − k k − k ≤ k − k≤ s 1 C 2   − k k Furthermore, for all 0 j n, ≤ ≤ √εj εn max R δ , R−1 δ . (2.9) | j,n − j,n| | j,n − j,n| ≤ 1 C 2   − k k

Proof. We first observe that, by Lemma 2.2 with k = n and (2.1), we have from (2.2)

n 2 2 2 2 Ce = ε = 1 f 2 = 1 R 1 R . (2.10) k nk n − k nkL (G) − | j,n| ≤ − | n,n| Xj=0 Hence R (0, 1] by the positivity of the leading coefficients in (2.1). This, in particular, n,n ∈ implies that 1 1 0 max(1 Rn,n, 1) Rn,n. (2.11) ≤ − Rn,n − ≤ Rn,n − In order to establish the norm estimates claimed above, we note that, by Corollary 2.3,

R−1 R∗ = (I R∗R)R−1 = C∗CR−1 − −

8 and hence, for any j 0, ≥ ε C 2 e∗(R−1 R∗) 2 = (Ce )∗CR−1 2 ε CR−1 2 jk k , k j − k k j k ≤ jk k ≤ 1 C 2 − k k where in the last inequality we have again applied Corollary 2.3. This yields the second inequality in (2.8). Taking into account (2.11) in conjunction with the obvious relation

∞ j−1 1 e∗(R−1 R∗) 2 = R−1 2 + R 2 + R 2, k j − k | j,n| |R − n,n| | n,j| n,n n=0 n=Xj+1 X we arrive at the first inequality in (2.8).

In order to obtain estimates for each entry of R−1 I and I R∗, we argue in terms of − − principal submatrices of order n + 1, and find upper bounds for the entries of the last row and and the last column of R−1 R∗ . We start with the last column. From Corollary 2.3 we n+1 − n+1 have M = Π∗ MΠ = R∗ R = I Π∗ C∗CΠ . Hence, n+1 n+1 n+1 n+1 n+1 − n+1 n+1 ∞ ℓ R−1 R∗ = (M −1 I)R∗ = Π∗ C∗CΠ R∗ n+1 − n+1 n+1 − n+1 n+1 n+1 n+1 Xℓ=1  = Π∗ C∗(I CΠ Π∗ C∗)−1CΠ R∗ . n+1 − n+1 n+1 n+1 n+1

Therefore, in view of the relation R∗ e = R e , we get for 0 j n that n+1 n n,n n ≤ ≤ √εjεn √εjεn R−1 δ R = e∗(R−1 R∗ )e R , (2.12) | j,n − j,n n,n| | j n+1 − n+1 n|≤ n,n 1 Π∗ C∗ 2 ≤ 1 C 2 − k n+1 k − k k which together with (2.11) gives the required estimate for the entries of R−1 and the diagonal entry of R.

Finally, for the non-diagonal entries we get, for j

R = e∗ (R−1 R∗ )e = e∗ Π∗ C∗CΠ R−1 e | j,n| | n n+1 − n+1 j| | n n+1 n+1 n+1 j| √ε CΠ R−1 e = √ε CΠ R−1 e = R √ε CΠ M −1 e ≤ nk n+1 n+1 jk nk j+1 j+1 jk j,j nk j+1 j+1 jk ∞ √εjεn √ε CΠ (Π∗ C∗CΠ )ℓe , ≤ n j+1 j+1 j+1 j ≤ 1 C 2 ℓ=0 − k k X as required to establish (2.9). 

Remark 2.5. With the above notations and estimates, we can produce an alternative proof of (1.7) and (1.8). First note that (1.4) and (2.1), together with (2.10), imply that

2n+2 2 n + 1 γ Rn,n = 2 (0, 1]. π λn ∈

−1 Hence, since Rn,n = 1/Rn,n, we obtain from (2.10) and (2.12) that 1 ε ε 1 R2 R n , n ≤ − n,n ≤ R − n,n ≤ 1 C 2 n,n − k k 9 which yields (1.7).

Now from (2.1) we have that

p (z) f (z) = (p (z), ..., p (z))(I R )e , n − n 0 n − n+1 n and thus, by (2.8),

n−1 2 2 2 2 2 εn C f R p 2 = R (I R )e = f p 2 k k , (2.13) k n − n,n nkL (G) | j,n| ≤ k − n+1 nk k n − nkL (G) ≤ 1 C 2 Xj=0 − k k implying (1.8). For κ-quasiconformal boundaries, recall that C κ. The relation k k≤ 2 2 εn λ f R p 2 k n − n,n nkL (G) ≤ 1 λ2 − has been established in [26, Theorem 2.1], where λ [0, 1) is defined by a quasiconformal ∈ reflection of Γ. 

2.1 The example of an ellipse

We conclude this section with a detailed study of the case of an ellipse. Here (1.5) is very simple, allowing to give explicitly the matrix C, and thus to illustrate many of the results presented so far.

Let G be the interior of the ellipse with semi-axes (r r−1)/2, for some r > 1. Then, the ± corresponding conformal map ψ : D∗ Ω is given by 7→ 1 1 ψ(w)= (rw + ), 2 rw and γ = 2/r. Clearly ψ has an analytic continuation for w > 0, which is however only | | univalent for w > ρ = 1/r (0, 1), as the equation ψ′(w) = 0 reveals. Hence Γ U(1/r) and | | ∈ ∈ ε = (r−2n) by (1.11). We show that, for ellipses, this estimate can be improved. n O Inserting the formula for ψ(w) in (1.1), we deduce the following formula for the associated Faber polynomials n −2n −n Fn(ψ(w)) = w + r w for n 1 and w D∗, see, e.g., [25]. Thus for w D∗ ≥ ∈ ∈ −2n−2 ′ n + 1 n n + 1 r rn(w)= ψ (w)fn(ψ(w)) w = n+2 . (2.14) − r π −r π w Comparing with (1.5) and (1.6) we conclude that

−2 −4 −2 2 2 −4n−4 C = diag(r , r , ...), C = r , ε = Ce = r 2 D∗ = r , (2.15) k k n k nk k nkL ( ) see, e.g., [15, 3]. It follows therefore from Corollary 2.3 that, like C, also M,R, and R−1 are § diagonal, with R = √1 ε . This leads to n,n − n ε 1 ε n e∗ (I R∗) = 1 R 1 n . 2 ≤ k n − k − n,n ≤ R − ≤ 1 C 2 n,n − k k 10 Comparing with Theorem 2.4, we see that, in case of an ellipse, the estimate (2.9) for j = n is sharp up to some constants, whereas (2.8) is not.

In view of (2.1), a diagonal R also implies the well-known fact [10, p. 547] that the n-th ′ Bergman polynomial for ellipses is proportional to Fn+1, and more precisely

n + 1 γ2n+2 √ fn(z)= Rn,npn(z),Rn,n = 1 εn = 2 . (2.16) − s π λn

Thus we obtain equality in (1.7), i.e.,

2n+2 n + 1 γ −4n−4 1 2 = εn = r , − π λn that is, (1.7) is sharp for analytic Γ. At the other hand, (1.8) and (1.9) are not sharp for ellipses since, in view of (2.16), we get for z Ω ∈

2 2 2 3 fn(z) fn pn L2(G) = (1 Rn,n) = εn/4+ (εn), = Rn,n =1+ (εn). k − k − O pn(z) O

−2n−2 Finally, it turns out that the exponential term √εn = r in our Theorem 1.1 cannot be improved for ellipses. Indeed, for z Ω we find using (2.14)–(2.16) that ∈ π p (z) π f (z) n 1 = n 1 n + 1 φ′(z)φ(z)n − n + 1 R φ′(z)φ(z)n − r r n,n 1 r−2n−2 = 1 2n+2 1 Rn,n − φ(z) −   √εn √εn = (εn + )= ( ) O φ(z) 2n O nβ/2 | | for all β > 0, uniformly on compact subsets K of Ω. Clearly, in the right-hand side we may not −2n−2 n −2 replace √εn = r by ρ for some ρ < r independent of K.

e e 3 Additional properties of the Grunsky matrix for smooth Γ

All asymptotic results obtained so far in this paper depend on whether ε = Ce 2 tends n k nk to zero, and with which speed. We have already seen in (1.11) three different examples of smoothness of Γ, for which we have further information on εn, cited from the work of Carleman, Suetin and one of the authors [3, 28, 26]. The aim of this section is to gather other known results in the literature about spectral properties of the Grunsky matrix, and relate them to the smoothness of Γ.

Before going further, it will be useful to recall the generating function for the Grunsky matrix, as well as some basic facts on compact operators. For w D∗ we define the infinite ∈ vector

ℓ + 1 1 1 y(w)= ℓ2, and hence y(w) 2 = (3.1) π wℓ+2 ℓ=0,1,... ∈ k k π( w 2 1)2 r  | | −

11 by elementary computations. The way that C depends on ψ is quite involved. To see it, for distinct w, v D∗, we differentiate (1.1) with respect to v and w and employ (1.2) to arrive at ∈ the generating function

′ ′ 1 ψ (w)ψ (v) 1 T = y(v) Cy(w)= Cy(w),y(v) 2 π (ψ(v) ψ(w))2 − (v w)2 − −h iℓ  − −  ∞ n + 1 r (w) = n , π vn+2 n=0 r X where in the last equality we made use of (1.5). It is well known (see, e.g., [23, pp. 2128–2129]) that for w = v we get a link with the Schwarzian derivative of ψ in Ω

ψ′′′(w) 3 ψ′′(w) 2 S (w) := = 6πy(w)T Cy(w), (3.2) ψ ψ′(w) − 2 ψ′(w) −   a well-known quantity to measure smoothness of ψ or Γ [19, 1.2.6 and 1.5]. In particular, § § using (3.1) we find that, for w D∗, ∈ ( w 2 1)2 y(w)T Cy(w) | | − S (w) = | | C . 6 | ψ | y(w) 2 ≤ k k k k

An operator B acting on ℓ2 is called compact if the image under B of any bounded sequence in ℓ2 contains a subsequence which is convergent in norm [13, III.4.1]. In particular, we deduce § that Be 0. We recall that B is compact if and only if the sequence of the singular values n → of B tends to 0 [13, V.2.3]. B is called compact of Schatten class of index p (trace class if § p = 1, Hilbert-Schmidt class if p = 2) if the sequence of singular values of B is an element of ℓp [13, X.1.3]. In particular B∗B is of trace class if and only if B is of Hilbert-Schmidt § class, i.e., ∞ Be 2 < . We also recall that the product of a compact operator with a n=0 k nk ∞ bounded operator is compact [13, Thm. III.4.8], and so is the adjoint [13, Thm. III.4.10]. Similar P properties hold for operators of Schatten class of index p [13, X.1.3]. Thus Corollary 2.3 and § Theorem 2.4 lead to the following.

Corollary 3.1. C is compact (or of p-Schatten class) if and only if I M and/or I M −1 are − − compact (or of p/2-Schatten class). Furthermore, if C is Hilbert-Schmidt then so is I R and − I R−1. − Remark 3.2. Suetin applies in [28, 1] the method of normal moments, where he requires that § ∞ e∗C∗Ce 2 < , | j k| ∞ j,kX=0 or, in operator terms, C∗C is of Hilbert-Schmidt class (what amounts to ask that C is of 4- Schatten class). To ensure this, he derives in [28, Lemma 1.5] and subsequent considerations an upper bound for e∗C∗Ce in terms of the smoothness properties of Γ, namely, he showed that | j k| there exists a constant c such that, for all j, k 0, ≥ c e∗C∗Ce , | j k|≤ (j + 1)β/2(k + 1)β/2 provided that Γ (p + 1, α) with p 0, and β = 2p + 2α > 1. For j = k = n this leads ∈ C ≥ to the estimate ε = Ce 2 = (n−β). As a consequence, Suetin only deals with C in the n k nk O Hilbert-Schmidt class, and thus I M = C∗C of trace class. We do not know whether Suetin −

12 was aware of the classical theory of determinants of trace class perturbations of the identity [6, 11.9]. Nevertheless, in [28, Lemma 1.6 and 1.7] this theory is used in order to compute the § asymptotic behavior or upper bounds for various determinants, in particular the existence of the following limit

n−1 2 2 lim R = lim det(Rn) = lim det(Mn) = det(M) > 0, n→∞ j,j n→∞ n→∞ jY=0 but also upper bounds for M e det n+1 n I (R−1 )∗e e∗ 0 R−1 R−1 = det n+1 n+1 n =  j  = e∗M −1 e . ± n,n j,n e∗R−1 0 det(M ) − j n+1 n  j n+1  n+1 

A compact Grunsky matrix C can be equivalently characterized by the fact that Γ is asymptotically conformal [23, Thm. 5.1], which by [19, Thm. 11.1] is equivalent to the property ( w 2 1)2S (w) 0 for w 1+. Recall from above that, in any of these cases, ε 0 for | | − ψ → | | → n → n . The converse is however wrong. →∞ Corollary 3.3. The following statements hold.

(a) If Γ has a corner (with angle different from π), then the corresponding Grunsky operator C is not compact. (b) For any compact Grunky operator we have that Γ is quasiconformal. (c) Γ is an analytic Jordan curve if and only if there exists ρ (0, 1) with ε = (ρ2n). ∈ n O

Proof. Part (a) follows by the observation that asymtotically conformal does not allow corners with angles different from π by [19, 11.2]. § To show part (b), recall from above that if the Grunsky operator is compact then Γ is asymptotically conformal. This latter property implies that Γ is also quasiconformal, see [19, 11.2]. § The implication = of part (c) has already been stated in (1.11). In order to show the ⇒ reciprocal, suppose that ε = (ρ2n) for some ρ (0, 1). Then there exists a constant c such n O ∈ that C = C Ce = √ε c ρℓ (3.3) | ℓ,0| | 0,ℓ| ≤ k ℓk ℓ ≤ for all ℓ 0. Thus we get from (1.4) and (1.5) for n = 0 that, for w > 1, ≥ | | ∞ ′ γ 1 ℓ + 1 −(ℓ+2) r0(w)= ψ (w) = w Cℓ,0, √π − √π − r π Xℓ=0 and, in view of (3.3), we conclude that ψ′ and thus ψ have an analytic continuation across the unit circle into D, and thus Γ is the analytic image of the unit circle. Therefore, around any w0 of modulus 1, the map ψ can be represented by a Taylor series expansion of the form ψ(w)= ψ(w )+ a (w w )+ a (w w )2 + .... 0 1 − 0 2 − 0

13 Γ analytic n ⇐⇒ εn = (ρ ) for some ρ (0, 1) (see [15, Eqn. (9)] and Corollary 3.3(c)) O ∈

⇓ 6⇑ Γ piecewise analytic without cusps = 2 ⇒ εn = Cen = (1/n) (see [26, Cor. 2.1]) k k O

⇓ 6⇑ Γ κ-quasiconformal for some κ [0, 1) C < 1 (with C κ, see [18, Theorem 9.12∈ and⇐⇒ Theorem 9.13]) k k k k≤ (no corner allowed, see [19, 11.2]) ⇑ 6⇓ § Γ asymptotically conformal 2 2 ⇐⇒ ( w 1) Sψ(w) 0 for w 1 (see [19, Thm. 11.1]) | | − C compact→ (see,| | e.g., → [23, Thm. 5.1]) ⇐⇒ = Cen 0 ⇒k k →

⇑ 6⇓ 2 2 2 D∗ ( w 1) Sψ(w) dA(w) < C is Hilbert-Schmidt| | − operator| | (see, e.g.,∞ [23, ⇐⇒ Thm. 5.2]) R ∞ 2 n=0 Cen < ⇐⇒ kβ k ∞ = εn = (1/n ) for some β > 1 =Γ (p +1, α) for⇐ some p OP0 and α (0, 1) such that β = 2(p + α) > 1 ⇐ ∈C (see≥ [28, Lemma∈ 1.5]).

Figure 1: Regularity of the boundary Γ, regularity of the Schwarzian derivative Sψ defined in (3.2), and properties of the Grunsky operator.

If a = ψ′(w ) = 0 then necessarily a = 0 since ψ is univalent in D∗. Thus w would be 1 0 2 6 0 mapped by ψ onto an exterior pointing cusp on Γ. Our assumption on εn also implies that C is of Hilbert-Schmidt class and thus is compact. From part (a) we conclude that Γ has no corners, and thus ψ′(w) = 0 for all w = 1. This implies univalence of ψ in a neighborhood of w = 1, 6 | | | | and thus Γ is an analytic Jordan curve. 

As an illustration of Corollary 3.3(a), if Γ is piecewise analytic without cusps, but with cor- ners different from π, we cannot have a compact Grunsky operator C though here ε = (1/n) n O by (1.11). Corollary 3.3(b) allows us to conclude that indeed Γ of class Γ(p+1, α) are included in the statements of 1 and 2 since they are quasiconformal. Finally, an assertion similar to Corol- § § lary 3.3(c) has been given before in [21, Theorem 2.4], but only with the additional condition that Γ has no zero interior angles.

We do not know whether the sole assumption of Γ being κ-quasiconformal (or C κ< 1) k k≤ is sufficient to imply that ε 0 for n . n → →∞

In Figure 1 we summarize the link between smoothness of Γ, boundedness of Sψ and spectral properties of C, with pointers to the literature.

14 It is not too difficult to see that the Grunsky matrix does not depend on a linear trans- formation of the plane. We may therefore suppose without loss of generality that 0 G, and ∈ that the inner conformal map ϕ : D G satisfies ϕ(0) = 0 and ϕ′(0) = 1 (in fact the outer 7→ conformal map ψ : D∗ 1/G is given by ψ(w) = 1/ϕ(1/w) with cap(1/Γ) = 1). We also notice 7→ that, if Γ is quasiconformal, then so is 1/Γ by [18, Lemma 9.8]. Several authors (see, e.g., [29, 2.2]) consider ane extension of C = C(Γ) ase a linear operator acting on ℓ2(Z), namely, § C B A = BT C   e which is complex symmetric, where C represents (up to permutation of rows and columns) the Grunsky operator C(1/Γ). Here a generalized Grunsky inequality shows that A is unitary (provided that G has positive planar Lebesguee measure and Γ has zero planar Lebesgue measure, which is true in our setting of a quasiconformal Jordan curve Γ). In particular, it is easily seen that M = I C∗C = B∗B. Jones in [12, Theorem 1.3] gives another criterion for C(1/Γ) to be − in Schatten class of index p in terms of the inner conformal map ϕ from D onto G. This, for p = 2, leads to the equivalent characterization that log ϕ′(w) 2dA(w) < . D | | ∞ R 4 Proofs of the main theorems

Proof of Theorem 1.1. Let K be a compact subset of Ω, and thus r = min φ(z) : z K > 1. {| | ∈ } Using (1.5) and (2.3) we have for z = ψ(w) K that ∈ π p (z) f (z) π ψ′(w) n − n = f (z), ..., f (z) R−1 I e n + 1 φ′(z)φ(z)n n + 1 wn 0 n n+1 − n+1 n r r   n  j π 1 −1 π j + 1 w −1 = n r0(w), ..., rℓ(w), ... R I en + n Rj,n δj,n . n + 1 w − n + 1 r π w − r    r Xj=0  

The right-hand side will be divided up in three terms, two of them being bounded above uniformly for z K by (√ε /rn/2), and the last one by (√ε /nβ/2), as this will be sufficient ∈ O n O n for the claim of Theorem 1.1. First notice that, by (1.5) and the use of the Cauchy-Schwarz inequality we have

π 1 r (w), ..., r (w), ... R−1 I e n + 1 wn 0 ℓ − n r    π 1 1 ℓ + 1 = w−2, ..., w−ℓ−2, ... CR −1 I R e n + 1 rn π π − n r r r    π 1 1 ℓ + 1 w−2, ..., w−ℓ−2, ... C R−1 (I R)e ≤ n + 1 rn π π k k k k − n r r r  −1 2 1 1 C R ε C εn/n k k k k n k k = ( ), ≤ n + 1 rn w 2 1 1 C 2 O rn r | | − s − k k p where in the last inequality we have applied (2.8) and (3.1).

Let now m be the integer part of (n + 1)/2. Then, again by the Cauchy-Schwarz inequality

15 and (2.8),

m−1 π j + 1 wj R−1 δ n + 1 π wn j,n − j,n r j=0 r   X 1 m + 1 = w−n( w0, ..., wm, 0, 0, ...)(R−1 I)e n + 1 n + 1 − n r r r−n (r0, ..., rm, 0, 0, ...) R−1 (I R)e ≤ k k k k k − nk m−n 2 r ε C √εn R−1 n k k = ( ). ≤ √r2 1 k k s1 C 2 O rn/2 − − k k

Now, using (2.9) we obtain for the remaining term that

n n π j + 1 wj ε 1 R−1 δ n (j + 1)ε rj−n. n + 1 π wn j,n − j,n ≤ n + 1 1 C 2 j r j=m r   r − k k j=m q X X

By our assumption on ε there exists some constant c such that ε c/(n + 1)β and thus, n n ≤ uniformly for m j n, ≤ ≤ c c 2βc ε . j ≤ (j + 1)β ≤ (m + 1)β ≤ (n + 1)β Hence,

n n ε 1 ε √2βc n (j + 1)ε rj−n n rj−n n + 1 1 C 2 j | ≤ (n + 1)β 1 C 2 r − k k jX=m q r − k k jX=m √ε = ( n ), O nβ/2 as claimed above.

Finally, by (1.13) and (1.14) we obtain

π f (z) ε /n n 1= ( n ) n + 1 φ′(z)φ(z)n − O rn r p uniformly for z K. Since r−n = (n−β) for all β > 0, the required relation (1.15) follows.  ∈ O

Proof of Theorem 1.2. We start by establishing a well-known formula for multiplication with z for our reference polynomials: Since f is of degree n, there exist G C, G = 0 for n j,n ∈ j,n j>n + 1, such that n+1 zfn(z)= fj(z)Gj,n. (4.1) Xj=0 It turns out that explicit formulas can be given for the entries Gj,n in terms of the conformal map ψ: We use the generating function [18, Section 3.1, Eqn.(4)]

∞ ∞ 1 F ′ (z) π f (z) = n+1 = n . (4.2) ψ(u) z (n + 1)un+1 n + 1 un+1 n=0 n=0 − X X r 16 Then, by multiplying by ψ(u) z and comparing like powers of u we conclude that, for j n+1, − ≤ n + 1 G = ψ . j,n j + 1 j−n r

For relating H with G, we denote by Hn, In, and Gn the submatrices of H, I, and G, respectively, formed with the first n + 1 rows and the first n columns. Then we may rewrite (1.16) as p (z), ..., p (z) (zI H ) = (0, ..., 0) 0 n n − n and obtain using (2.1) and (2.3) 

(0, ..., 0) = p (z), ..., p (z) (zI H )R 0 n n − n n   = f (z), ..., f (z) R−1 (zI H )R 0 n n+1 n − n n   = f (z), ..., f (z) (zI R−1 H R ). 0 n n − n+1 n n  

By the uniqueness of the coefficients in (4.1) we conclude that Rn+1Gn = HnRn for all n. Since below Gn in G and below Rn in R all entries are equal to zero, we conclude that

RG = HR, (4.3) which together with Corollary 2.3 and the boundedness of H (see (1.16)) implies that G repre- sents a bounded operator. Furthermore, for all 1 k n and n 0, − ≤ ≤ ≥ H G = e∗ H G e n−k,n − n−k,n n−k − n   = e∗ H(I R) (I R)G e n−k − − − n n n+1 = H (I R) (I R) G . n−k,ℓ − ℓ,n − − n−k,ℓ ℓ,n ℓ=nX−k−1 ℓ=Xn−k

Using (2.9), we get the upper bound

n n+1 1 H G H √ε ε + G √ε ε , | n−k,n − n−k,n|≤ 1 C 2 k k ℓ n k k ℓ n−k − k k  ℓ=nX−k−1 ℓ=Xn−k  and the claim in Theorem 1.2 follows. 

Acknowledgements. Parts of this work have been carried out during a visit of B.B. at the University of Cyprus in October 2014 and of N.S. at the University of Lille in July 2015. Both authors thank the hosting universities for their support.

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