#  An extremal problem for the Bergman kernel
of orthogonal polynomials

S. Charpentier, N. Levenberg and F. Wielonsky

###### Abstract

Let $\Gamma\subset\mathbb{C}$ be a curve of class $C(2,\alpha)$. For $z_{0}$
in the unbounded component of $\mathbb{C}\setminus\Gamma$, and for
$n=1,2,...$, let $\nu_{n}$ be a probability measure with
$\mathop{\mathrm{supp}}\nolimits(\nu_{n})\subset\Gamma$ which minimizes the
Bergman function $B_{n}(\nu,z):=\sum_{k=0}^{n}|q_{k}^{\nu}(z)|^{2}$ at $z_{0}$
among all probability measures $\nu$ on $\Gamma$ (here,
$\\{q_{0}^{\nu},\ldots,q_{n}^{\nu}\\}$ are an orthonormal basis in
$L^{2}(\nu)$ for the holomorphic polynomials of degree at most $n$). We show
that $\\{\nu_{n}\\}_{n}$ tends weak-* to $\widehat{\delta}_{z_{0}}$, the
balayage of the point mass at $z_{0}$ onto $\Gamma$, by relating this to an
optimization problem for probability measures on the unit circle. Our proof
makes use of estimates for Faber polynomials associated to $\Gamma$.

##  1 Introduction

Let $K$ be a compact subset of the complex plane $\mathbb{C}$ and let
$\mathcal{M}(K)$ denote the probability measures on $K$. Given a positive
integer $n$, if the support of $\nu\in\mathcal{M}(K)$ contains at least $n+1$
points, we can form the associated Bergman function
$B_{n}(\nu,z):=\sum_{k=0}^{n}|q_{k}^{\nu}(z)|^{2}$ where
$\\{q_{0}^{\nu},\ldots,q_{n}^{\nu}\\}$ form an orthonormal basis in
$L^{2}(\nu)$ for $\mathcal{P}_{n}$, the holomorphic polynomials of degree at
most $n$. We fix $z_{0}\in\Omega$, the unbounded component of
$\mathbb{C}\setminus K$, and for each $n$ we consider a probability measure
$\nu_{n}$ with $\mathop{\mathrm{supp}}\nolimits(\nu_{n})\subset K$ which
minimizes the Bergman function at $z_{0}$ among all such
$\nu\in\mathcal{M}(K)$:

$B_{n}(\nu_{n},z_{0})=\min_{\nu\in\mathcal{M}(K)}B_{n}(\nu,z_{0}).$

Equivalently, $\nu_{n}$ solves the max-min problem

$\max_{\nu\in\mathcal{M}(K)}\lambda_{n}(\nu,z_{0}),\qquad\lambda_{n}(\nu,z_{0})=\min_{p\in\mathcal{P}_{n},~{}p(z_{0})=1}\int_{K}|p|^{2}d\nu\leq
1,$ (1.1)

where $\lambda_{n}(\nu,z_{0})$ is the Christoffel function of $\nu$ at
$z_{0}$. We recall that (cf., [11, Theorem 1.4])

$\lambda_{n}(\nu,z)=B_{n}(\nu,z)^{-1},\qquad n\geq 0.$

Such an extremal measure $\nu_{n}$ is called an optimal prediction measure
(OPM) for $K$ and $z_{0}$ of order $n$. In general, it is not unique. For
motivation to study this problem, we refer to [3] where they give a nice
application to the field of optimal design for polynomial regression. Although
$B_{n}(\nu,z)$ is well defined only if all orthogonal polynomials up to degree
$n$, exist, $\lambda_{n}(\nu,z)$ is always defined, equal to $0$ when the
support of $\nu$ consists of fewer than $n+1$ points. In fact,
$\lambda_{n}(\nu,z)$ is defined for all $z\in\mathbb{C}$. For an extremal
measure $\nu_{n}$, all the orthogonal polynomials $q_{k}^{\nu_{n}}$,
$k=0,\ldots,n$, do exist. Note also that, for each $n$, the Bergman function
$B_{n}(\nu,z)$ only depends on a finite number of moments of the measure
$\nu$, namely

$m_{j,k}=\int_{K}z^{j}\overline{z}^{k}d\nu,\quad j,k=0,\ldots,n.$ (1.2)

It is known that $B_{n}(\nu_{n},z_{0})$ is related to the polynomial of
extremal growth at $z_{0}$, see [3]. Indeed, one has

$B_{n}(\nu_{n},z_{0})=\sup_{p\in\mathcal{P}_{n}}\frac{|p(z_{0})|^{2}}{\|p\|_{K}^{2}}\leq
e^{2ng_{\Omega}(z_{0})},$ (1.3)

where the upper bound, with $g_{\Omega}$ the Green function of $\Omega$,
follows from the fact that

$g_{\Omega}(z_{0})=\sup\\{\frac{1}{deg(p)}\log|p(z_{0})|:\
p\in\cup_{n}\mathcal{P}_{n},\ ||p||_{K}\leq 1\\}.$ (1.4)

Here deg$(p)$ denotes the degree of $p$ and $||p||_{K}:=\sup_{z\in K}|p(z)|$.
For a general probability measure $\nu$ on $K$ and $z\in\mathbb{C}$, we have
that

$1\geq\lambda_{n}(\nu,z)\geq\lambda_{n+1}(\nu,z)\geq 0$

so that the limit

$\lambda_{\infty}(\nu,z):=\lim_{n\to\infty}\lambda_{n}(\nu,z)$ (1.5)

exists and $0\leq\lambda_{\infty}(\nu,z)\leq 1$. It has been verified by
explicit computations in [3] that if:
i) $K=[-1,1]$ and $z_{0}$ is real or purely imaginary,
ii) $K=\overline{\mathbb{D}}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$ and $|z_{0}|>1$,
certain sequences of optimal prediction measures $\nu_{n}$ tend weak-* to a
limit, namely

$\nu_{n}\to\widehat{\delta}_{z_{0}},\qquad n\to\infty,$ (1.6)

where $\widehat{\delta}_{z_{0}}$ denotes the balayage measure of
$\delta_{z_{0}}$, the point mass at $z_{0}$, onto $K$. The authors of [3] have
conjectured that this convergence holds true more generally. It is the aim of
the present paper to show that the conjecture holds true for a more general
class of compact sets $K$ and points $z_{0}$ outside of $K$. Namely, our main
result is the following theorem.

######  Theorem 1.1.

Assume $K$ is a compact subset bounded by a curve $\Gamma\in C(2,\alpha)$,
$0<\alpha<1$ (i.e. $\Gamma$ can be parameterized by a function of class
$C(2,\alpha)$). For $z_{0}\in\Omega$, any sequence of optimal prediction
measures $\\{\nu_{n}\\}_{n}$ tends weak-* to $\widehat{\delta}_{z_{0}}$, the
balayage of $\delta_{z_{0}}$ onto $\Gamma$.

After some general preliminaries in the next section, in section 3 we
complement the study in [3] of the case of $K=\overline{\mathbb{D}}$, the
closed unit disk. We show in Theorem 3.2 that for $z\in\mathbb{D}$, the
balayage $\widehat{\delta}_{z}$ to $\mathbb{T}:=\partial\mathbb{D}$ is the
unique maximizer of $\lambda_{\infty}(\mu,z)$ from (1.5) among
$\mu\in\mathcal{M}(\mathbb{T})$. We then study the more general case of $K$
bounded by a real analytic curve $\Gamma$ in section 4. To derive Theorem 1.1
in this setting, for $z\in\Omega$ we make a connection between
$\widetilde{\lambda}_{\infty}(\nu,z)$, a modification of $\lambda_{\infty}$
for measures $\nu$ supported on $\Gamma$, with
$\lambda_{\infty}(\Phi_{*}\nu,1/\overline{\Phi(z)})$ where $\Phi_{*}\nu$ is
the push-forward of $\nu$ on $\mathbb{T}$, $\Phi$ being a conformal map from
the exterior of $\Gamma$ to the exterior of $\mathbb{T}$. The extension to the
case $\Gamma\in C(2,\alpha)$ is then easily obtained by appealing to a
classical estimate on Faber polynomials. An outline of the proof is given at
the beginning of section 4.2, followed by the details. We conclude with an
interesting observation on the distinction between the cases of $K$ being a
curve versus $K$ being an arc.

##  2 General preliminaries

We begin with some observations related to [3].

  1. 1.

The max-min in (1.1) coincides with the min-max for general compact $K$,
namely

$\max_{\nu\in\mathcal{M}(K)}\min_{p\in\mathcal{P}_{n},~{}p(z_{0})=1}\int_{K}|p|^{2}d\nu=\min_{p\in\mathcal{P}_{n},~{}p(z_{0})=1}\max_{\nu\in\mathcal{M}(K)}\int_{K}|p|^{2}d\nu.$
(2.1)

This follows from the classical minimax theorem, see the proof of [3,
Proposition 2.1].

  2. 2.

Let $K\subset\mathbb{C}$ be compact and contain infinitely many points and fix
$z_{0}\not\in K$. For $n\in\mathbb{N}$, let

$M_{n}=M_{n}(z_{0},K):=\sup\\{|p(z_{0})|:p\in\mathcal{P}_{n},\ ||p||_{K}\leq
1\\}.$ (2.2)

There exists a unique $p_{n}\in\mathcal{P}_{n}$ with $||p_{n}||_{K}=1$ and
$p_{n}(z_{0})=M_{n}$; in [3] this is called the polynomial of extremal growth
relative to $K$ at $z_{0}$. Indeed, note that

$M_{n}=\sup\\{\frac{|p(z_{0})|}{||p||_{K}}:p\in\mathcal{P}_{n}\\}=[\inf\\{\frac{||p||_{K}}{|p(z_{0})|}:p\in\mathcal{P}_{n}\\}]^{-1}$

and

$\displaystyle\inf\\{\frac{||p||_{K}}{|p(z_{0})|}:p\in\mathcal{P}_{n}\\}$
$\displaystyle=\inf\\{||p||_{K}:p\in\mathcal{P}_{n},\ |p(z_{0})|=1\\}$
$\displaystyle=\inf\\{||1-Q||_{K}:Q\in\mathcal{P}_{n},\ Q(z_{0})=0\\}.$

Let $V_{n}:=\\{Q\in\mathcal{P}_{n},\ Q(z_{0})=0\\}$. This is an
$n-$dimensional complex vector space, and clearly each nonzero $Q\in V_{n}$
has at most $n-1$ zeros in $K$ (since $Q(z_{0})=0$). By the classical Haar
uniqueness theorem in Chebyshev approximation (cf., [1], [5, Theorem 19]),
every continuous, complex-valued function on $K$ admits a unique best sup-norm
approximant from $V_{n}$. Applying this to the constant function $1$ there
exists a unique $Q_{n}\in V_{n}$ with $M_{n}=[||1-Q_{n}||_{K}]^{-1}$, and thus
$p_{n}=1-Q_{n}$.

  3. 3.

From 2. and Remark 2.3 in [3], it follows that the support of an OPM $\nu_{n}$
of order $n$ for $K$ and $z_{0}$ as in 2. is contained in

$S_{n}(K):=\\{z\in K:|p_{n}(z)|=||p_{n}||_{K}\\}.$

The set $\\{z\in\mathbb{C}:|p_{n}(z)|=||p_{n}||_{K}\\}$ is a real algebraic
curve in $\mathbb{R}^{2}$ of degree at most $2n$. In particular, for
$z_{0}\in\Omega$, the unbounded component of $\mathbb{C}\setminus K$, if
$p_{n}$ is non-constant, any OPM $\nu_{n}$ for $K$ is supported on
$\partial\Omega$. A necessary and sufficient condition that $p_{n}$ be non-
constant is that $z_{0}$ lie outside of

$\widehat{K}_{n}:=\\{z\in\mathbb{C}:|q_{n}(z)|\leq||q_{n}||_{K}\ \hbox{for
all}\ q_{n}\in\mathcal{P}_{n}\\},$

the $n-$th order polynomial hull of $K$. Since these sets $\widehat{K}_{n}$
decrease to

$\widehat{K}:=\\{z\in\mathbb{C}:|q(z)|\leq||q||_{K}\ \hbox{for all}\
q\in\bigcup_{n}\mathcal{P}_{n}\\},$

the polynomial hull of $K$, and $\Omega=\mathbb{C}\setminus\widehat{K}$, by
appealing to either the Hilbert lemniscate theorem (cf.,[7, Theorem 5.5.8]) or
simply Runge’s theorem, for any $z_{0}\in\Omega$, there exists $n_{0}$
sufficiently large so that $p_{n}$ is non-constant for $n\geq n_{0}$.

Thus if, e.g., $K$ is an ellipse of the form

$K=\\{(x,y)\in\mathbb{R}^{2}:x^{2}/a^{2}+y^{2}/b^{2}=1\\}$

with $a\not=b$ and $z_{0}$ lies outside $K$, by Bezout’s theorem $S_{n}(K)$
contains at most $4n$ points. Since an OPM $\nu_{n}$ exists, the support of
$\nu_{n}$ contains at least $n+1$ points. On the other hand, we recall in the
next section that for the unit circle
$\mathbb{T}=\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1\\}$ and a point $z_{0}$
with $|z_{0}|>1$, there exist OPM’s $\nu_{n}$ which are absolutely continuous
with respect to arclength measure and hence with support $\mathbb{T}$. It
follows from 3. that OPM’s for $\overline{\mathbb{D}}$ and $\mathbb{T}$
coincide. More generally, if $K$ is a compact subset bounded by an analytic
curve $\Gamma=\partial\Omega$ (as in section 4) and $z_{0}\in\Omega$, OPM’s
$\nu_{n}$ for $K$ and $\Gamma$ coincide, at least for $n$ sufficiently large,
which we will always assume.

##  3 The unit disk $\mathbb{D}$

We begin by recalling that the harmonic measure $\omega_{\mathbb{D}}(z,t)$ for
the disk $\mathbb{D}$ and a point $z=|z|e^{i\phi}\in\mathbb{D}$ is given by

$d\omega_{\mathbb{D}}(z,t)=\frac{1-|z|^{2}}{|e^{it}-z|^{2}}\frac{dt}{2\pi}=\left[\sum_{k=-\infty}^{\infty}\left|z\right|^{|k|}e^{ik(\phi-t)}\right]\frac{dt}{2\pi},$
(3.1)

see e.g. [7, Chapter 4.3]. It may also be defined as the balayage
$\widehat{\delta}_{z}$ of the Dirac mass $\delta_{z}$ onto the unit circle
$\mathbb{T}$, see [8, Appendix A.3] or, by conformal invariance, the balayage
$\widehat{\delta}_{1/\overline{z}}$ of $\delta_{1/\overline{z}}$ onto
$\mathbb{T}$.

######  Definition 3.1.

A positive measure $\mu$ on the unit circle $\mathbb{T}$ satisfies the Szegő
condition if its density $f=d\mu/d\theta$ satisfies

$\int_{\mathbb{T}}\log fd\theta>-\infty.$

Then, the Szegő function is defined by

$D(\mu,z)=\exp\left(\frac{1}{4\pi}\int_{0}^{2\pi}\frac{e^{it}+z}{e^{it}-z}\log
f(t)dt\right),\qquad|z|<1.$ (3.2)

Note that, with $\mu_{a}$ the absolutely continuous part of $\mu$, and
$\lambda>0$, one has

$D(\mu,z)=D(\mu_{a},z),\qquad D(\lambda\mu,z)=\sqrt{\lambda}D(\mu,z).$ (3.3)

It is known, see [9, Theorem 2.4.1], that for any measure $\mu$ satisfying the
Szegő condition,

$\lambda_{\infty}({\mu},z)=(1-|z|^{2})|D(\mu,z)|^{2},\qquad|z|<1.$ (3.4)

We also recall that for any measure $\mu$ on $\mathbb{T}$ the Christoffel
function satisfies

$|z|^{2n}\lambda_{n}({\mu},z)=\lambda_{n}({\mu},1/\overline{z}),\qquad z\neq
0,$ (3.5)

see e.g. [9, Lemma 2.2.8]. These relations (3.4) and (3.5) will be crucial in
the sequel, as will the unicity in the next result.

######  Theorem 3.2.

Let $z\in\mathbb{D}$. The unique probability measure $\mu$ on $\mathbb{T}$
that maximizes $\lambda_{\infty}({\mu},z)$, is the balayage measure
$\widehat{\delta}_{z}$.

###### Proof.

Let $\mu$ be a probability measure on $\mathbb{T}$. By [9, Theorem 2.7.15], we
have $\lambda_{\infty}({\mu},z)=0$ for any $z\in\mathbb{D}$ precisely when
$\mu$ does not belong to the Szegő class. Thus, we may assume that $\mu$
belongs to the Szegő class. By (3.4), we are led to maximize $|D(\mu,z)|$. Let
$\mu=\mu_{a}+\mu_{s}$, $\mu_{a}=fdt$, $f\in L^{1}(\mathbb{T})$, be the Radon-
Nikodym decomposition of the measure $\mu$. From (3.3), we see that the larger
the mass of $\mu_{a}$, the larger the modulus of $D(\mu,z)$. Thus, $\mu_{s}$
should vanish, that is, $\mu$ has to be absolutely continuous. Hence, we
assume that $\mu=fdt$, $f\in L^{1}(\mathbb{T})$. We want to maximize

$F(z):=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log f(e^{it}){\rm
Re}\,\left(\frac{e^{it}+z}{e^{it}-z}\right)dt$

over positive functions $f$ such that

$f\in L^{1}(\mathbb{T}),\quad\log f\in
L^{1}(\mathbb{T}),\quad\int_{-\pi}^{\pi}f(e^{it})dt=1.$ (3.6)

Let us consider the Möbius transform

$\varphi(u)=\frac{u+z}{1+\overline{z}u},$

so that $\varphi({0})=z$, and set $G=F\circ\varphi$. We have to maximize
$G(0)$. Since the boundary values of $G$ on $\mathbb{T}$ are
$\log(f\circ\varphi)$, it is equivalent to maximize

$\int_{-\pi}^{\pi}\log(f\circ\varphi)(e^{it})dt$ (3.7)

over functions $f$ satisfying the conditions (3.6). A computation shows that

$\int_{-\pi}^{\pi}\frac{(f\circ\varphi)(e^{it})}{|e^{it}+z|^{2}}dt=\frac{1}{1-|z|^{2}}\int_{-\pi}^{\pi}f(e^{it})dt.$

Hence, by considering the function

$g(e^{it}):=\pi(f\circ\varphi)(e^{it})\frac{1-|z|^{2}}{|e^{it}+z|^{2}},$

we see that maximizing (3.7) is equivalent to maximizing

$\int_{-\pi}^{\pi}\log g(e^{it})dt$

over positive functions $g$ such that

$\int_{-\pi}^{\pi}g(e^{it})\frac{dt}{2\pi}=1/2.$

By Jensen’s inequality,

$\int_{-\pi}^{\pi}\log
g(e^{it})\frac{dt}{2\pi}\leq\log\int_{-\pi}^{\pi}g(e^{it})\frac{dt}{2\pi}=-\log
2,$

and equality occurs only when $g$ is a constant function. Consequently, the
function $f$ has to be chosen so that
$(f\circ\varphi)(e^{it})=c_{1}|e^{it}+z|^{2}$ where $c_{1}=c_{1}(z)$ is a
constant depending on $z$, or equivalently,

$f(e^{it})=c_{2}|e^{it}-z|^{-2},\ c_{2}=c_{2}(z).$

By the normalization of $f$ in (3.6), we must actually have

$f(e^{it})=\frac{1}{2\pi}\frac{1-|z|^{2}}{|e^{it}-z|^{2}},$

which is the density of $\widehat{\delta}_{z}$. ∎

It is proved in [3] that, for a given degree $n$ and $z_{0}=|z_{0}|e^{i\phi}$
with $|z_{0}|>1$, the harmonic measure (3.1) for $1/\overline{z}_{0}$,

$\mathrm{d}\mu_{P}(\theta):=d\omega_{\mathbb{D}}(1/\overline{z}_{0},\theta)=\left[\sum_{k=-\infty}^{\infty}\left|z_{0}\right|^{-|k|}e^{ik(\phi-\theta)}\right]\frac{\mathrm{d}\theta}{2\pi},$

is an OPM of order $n$ for $\overline{\mathbb{D}}$ and $z_{0}$, as well as any
measure $\mu$ whose moments

$m_{k}=m_{k}(\mu):=\int_{\mathbb{T}}z^{k}d\mu,\qquad k=0,\pm 1,\ldots,\pm n$

coincide with those of the harmonic measure:

$m_{k}(\mu_{P}):=\int_{\mathbb{T}}z^{k}d\mu_{P}=\int_{\mathbb{T}}z^{k}d\widehat{\delta}_{1/\overline{z}_{0}}=\begin{cases}(\overline{z}_{0})^{-k},&k\geq
0,\\\\[5.0pt] z_{0}^{k},&k<0,\end{cases}\qquad|k|=0,1,\ldots,n.$

Moreover, from (1.3) and (2.2), since $M_{n}(z_{0},\mathbb{T})=|z_{0}|^{n}$ we
have, for $n\geq 0$,

$B_{n}(\mu_{P},z_{0})=|z_{0}|^{2n},\qquad\lambda_{n}(\mu_{P},z_{0})=|z_{0}|^{-2n}.$
(3.8)

##  4 $K$ bounded by a real analytic curve $\Gamma$

###  4.1 Preliminaries for $K$ bounded by a real analytic curve $\Gamma$

Let $K$ be a connected, simply connected, compact subset of $\mathbb{C}$ and
let $\Omega$ be the unbounded component of $\mathbb{C}\setminus K$. In this
section, we assume that $\Gamma=\partial\Omega$ is a real analytic curve. Let
$\Phi$ be a conformal map from $\Omega$ to $\mathbb{C}\setminus\mathbb{D}$.
Since $\Gamma$ is analytic, the map $\Phi$ can be analytically continued in a
neighborhood of $\Gamma$, inside $K$. Thus, there exists some $\epsilon>0$
such that, for all $r\geq 1-\epsilon$, the level curves of $\Phi$,

$\Gamma_{r}:=\\{z\in\Omega,~{}|\Phi(z)|=r\\},$

are well-defined. We will also denote by $\Omega_{r}$ the unbounded component
of $\mathbb{C}\setminus\Gamma_{r}$.

Moreover, given $R>1$, there exists a constant $m(\Phi)>0$ such that
$|\Phi^{\prime}(z)|\leq m(\Phi)$ for all $z$ such that $R\geq|\Phi(z)|\geq
1-\epsilon$. Hence for $1-\epsilon\leq r<R$ we have

$\mbox{{\rm dist}}(\Gamma_{R},\Gamma_{r})\geq m(\Phi)^{-1}(R-r).$ (4.1)

We will make use of the Faber polynomials $F_{n}$, $n\geq 0$, of the interior
$U$ of $K$, see [10]. Recall that they satisfy the following identity, see
[10, p.62]:

$F_{n}(z)=\Phi^{n}(z)+\frac{1}{2\pi
i}\int_{\Gamma_{r}}\frac{\Phi^{n}(t)}{t-z}dt,\qquad|\Phi(z)|>r.$

Note that, since $\Gamma$ is analytic and $\Phi$ can be continued in
$\Omega_{1-\epsilon}$, we can assume $r\geq 1-\epsilon$ in the above equality.
It follows that, for $1-\epsilon<r<R$, and for $z\in\Gamma_{R}$,

$\displaystyle F_{n}(z)$
$\displaystyle=\Phi^{n}(z)\left(1+\mathcal{O}\left(\frac{r^{n}}{\mbox{{\rm
dist}}(\Gamma_{R},\Gamma_{r})R^{n}}\right)\right)$
$\displaystyle=\Phi^{n}(z)\left(1+\mathcal{O}\left(\frac{r^{n}}{(R-r)R^{n}}\right)\right),\qquad
n\to\infty,$ (4.2)

uniformly on $\Gamma_{R}$, where the last equality uses (4.1). Moreover, it is
known that for $r>1$, there exists $n_{0}$ so that for $n>n_{0}$, all zeros of
$F_{n}$ lie inside $\Gamma_{r}$ [10].

We set $\Omega_{\infty}:=\Omega\cup\\{\infty\\}$. In the next subsection, we
will use the following proposition.

######  Proposition 4.1 ([2, Theorem 1]).

Let $\\{Q_{n}\\}_{n\geq 0}$, $\mbox{{\rm deg} }Q_{n}=n$, be a sequence of
monic polynomials with no zeros in $\Omega$. Then, the following three
statements are equivalent:
1) $\lim_{n\to\infty}\|Q_{n}\|_{K}^{1/n}=\mbox{{\rm cap}\,}(K)$, the
logarithmic capacity of $K$.
2) $\lim_{n\to\infty}|Q_{n}(z)|^{1/n}=\mbox{{\rm cap}\,}(K)e^{g_{\Omega}(z)}$,
locally uniformly in $\Omega$.
3) For any function $f$ analytic in $\Omega_{\infty}$, with a Taylor expansion
at infinity,

$f(z)=c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots,$

the sequence of Padé type approximants $P_{n}/Q_{n}$ satisfying

$Q_{n}(z)f(z)-P_{n}(z)=\mathcal{O}(z^{-1}),\quad\text{as }z\to\infty,$

converges locally uniformly to $f$ in $\Omega_{\infty}$.

This result remains true for any regular compact set $K\subset\mathbb{C}$.

###  4.2 Case of $K$ bounded by a real analytic curve $\Gamma$

Let $\mu$ be a probability measure on $\Gamma$. We set, for $z\in\Omega$,

$\widetilde{B}_{n}(\mu,z):=\frac{B_{n}(\mu,z)}{|\Phi(z)|^{2n}}\quad\hbox{and}\quad\widetilde{\lambda}_{n}(\mu,z):=|\Phi(z)|^{2n}\lambda_{n}(\mu,z),\qquad
n\geq 0,$ (4.3)

and

$\widetilde{B}_{\infty}(\mu,z):=\limsup_{n\to\infty}\widetilde{B}_{n}(\mu,z)\leq\infty\quad\hbox{and}\quad\widetilde{\lambda}_{\infty}(\mu,z):=\liminf_{n\to\infty}\widetilde{\lambda}_{n}(\mu,z)\geq
0.$ (4.4)

In fact, in Lemma 4.2 below, we show that the limits exist in (4.4).

The idea behind our proof that the weak-* limit of any subsequence
$\\{\nu_{n}\\}_{n\in Y},\ Y\subset\mathbb{N}$ of OPM’s for $\Gamma$ and $z$ is
$\widehat{\delta}_{z}$, the balayage of the point mass at $z$ to $\Gamma$, is
as follows. Using Proposition 4.1, we first show in Lemma 4.2 and Corollary
4.3 that for any probability measure $\mu$ on $\Gamma$,
$\widetilde{\lambda}_{\infty}(\mu,z)$ (and hence
$\widetilde{B}_{\infty}(\mu,z)$) is related to
$\lambda_{\infty}(\Phi_{*}\mu,1/\overline{\Phi(z)})$ where
$\Phi_{*}\mu\in\mathcal{M}(\mathbb{T})$, $\Phi$ being a conformal map from the
exterior of $\Gamma$ to the exterior of $\mathbb{T}$. The crux of the matter
is to then show that if $\alpha$ is a weak-* limit of a subsequence
$\\{\nu_{n}\\}_{n\in Y_{1}},\ Y_{1}\subset Y$ then (a perturbation of) a
“diagonal subsequence” $\\{\widetilde{B}_{n}(\nu_{n},z)\\}_{n\in Y_{1}}$,
converges to (a perturbation of) $\widetilde{B}_{\infty}(\alpha,z)$ (Lemma
4.7). As in the proof of Lemma 4.2, we use Faber polynomials in Lemma 4.4 as a
tool to prove a sort of monotonicity of $\\{\widetilde{B}_{n}(\mu,z)\\}$ in
$n$ for general $\mu$ which is needed to apply Dini’s theorem to conclude the
proof of Lemma 4.7. After the proof of our main result, we make a remark to
indicate a relationship with an underlying general potential-theoretic
question.

We denote by $A_{e}(\Gamma)$ the set of functions continuous on
$\overline{\Omega}_{\infty}$ which are analytic on $\Omega_{\infty}$.

######  Lemma 4.2.

Let $z\in\Omega$ and let $\mu$ be a measure on $\Gamma$. We have

$\widetilde{\lambda}_{\infty}(\mu,z)=\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}.$ (4.5)

Moreover,
$\widetilde{\lambda}_{\infty}(\mu,z)=\lim_{n\to\infty}\widetilde{\lambda}_{n}(\mu,z)$,
i.e., the limit exists.

###### Proof.

We have, for each $n$,

$\displaystyle\widetilde{\lambda}_{n}(\mu,z)$
$\displaystyle=|\Phi(z)|^{2n}\inf\left\\{\int|p|^{2}d\mu,~{}p\in\mathcal{P}_{n},~{}p(z)=1\right\\}$
$\displaystyle=|\Phi(z)|^{2n}\inf\left\\{\int\frac{|p|^{2}}{|\Phi|^{2n}}d\mu,~{}p\in\mathcal{P}_{n},~{}p(z)=1\right\\}$
$\displaystyle=\inf\left\\{\int\frac{|p|^{2}}{|\Phi|^{2n}}d\mu,~{}p\in\mathcal{P}_{n},~{}p(z)=|\Phi(z)|^{n}\right\\}$
$\displaystyle\geq\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}.$

Hence

$\liminf_{n\to\infty}\widetilde{\lambda}_{n}(\mu,z)=\widetilde{\lambda}_{\infty}(\mu,z)\geq\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}.$

To show that

$\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}\geq\limsup_{n\to\infty}\widetilde{\lambda}_{n}(\mu,z),$

given $\epsilon>0$, take $g\in A_{e}(\Gamma)$ with $g(z)=1$ and

$\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}\geq\int|g|^{2}d\mu-\epsilon.$

We show that for $n\geq n_{0}(\epsilon)$, we can find
$p_{n}\in\mathcal{P}_{n}$ such that

$\int|g|^{2}d\mu\geq\int\frac{|p_{n}|^{2}}{|\Phi|^{2n}}d\mu-\epsilon\quad\hbox{and}\quad
p_{n}(z)=|\Phi(z)|^{n}.$ (4.6)

To this end, for $r>1$, let

$J_{r}(t):=\Phi^{-1}(r\Phi(t)),\qquad\mu_{r}:=J_{r*}\mu.$

The measure $\mu_{r}$ is supported on $\Gamma_{r}$, and since $g\circ J_{r}$
tends uniformly to $g$ on $\Gamma$ as $r$ tends to 1, we have

$\int|g|^{2}d\mu_{r}=\int|g\circ J_{r}|^{2}d\mu\to\int|g|^{2}d\mu,\qquad r\to
1.$

Hence it suffices to approximate $\int|g|^{2}d\mu_{r}$.

Consider the sequence of Faber polynomials $\\{F_{n}\\}_{n\geq 0}$ of the
domain $U$. They satisfy the second item in Proposition 4.1 (up to the
constant factor $\mbox{{\rm cap}\,}(K)$), and for $n$ large, their zeros will
lie inside $\Gamma_{r/2}$. Hence, Proposition 4.1 can be applied outside
$\Gamma_{r/2}$, which shows that, for all $n$ sufficiently large, there exists
a polynomial $P_{n}$, of degree at most $n$, such that

$P_{n}/F_{n}\to g,~{}\text{uniformly in }\Gamma_{r},\ \hbox{and}\
P_{n}/F_{n}(z)\to 1.$

Moreover,

$F_{n}\to\Phi^{n},~{}\text{uniformly in }\Gamma_{r},\ \hbox{and}\
F_{n}(z)\to\Phi^{n}(z).$

Then for $n\geq n_{0}(\epsilon)$, for appropriate $c_{n}$ with $|c_{n}|\to 1$,
$p_{n}:=c_{n}P_{n}/P_{n}(z)$ satisfy (4.6). ∎

For a measure $\mu$ on $\Gamma$ we have $\Phi_{*}\mu$ is a measure on the
circle $\mathbb{T}$. From (1.5),

$\lambda_{\infty}(\Phi_{*}\mu,1/\overline{\Phi(z)}):=\lim_{n\to\infty}\lambda_{n}(\Phi_{*}\mu,1/\overline{\Phi(z)}).$

######  Corollary 4.3.

For any measure $\mu$ on $\Gamma$, it holds that

$\widetilde{\lambda}_{\infty}(\mu,z)=\lambda_{\infty}(\Phi_{*}\mu,1/\overline{\Phi(z)}),\qquad
z\in\Omega.$ (4.7)

###### Proof.

One has, in view of (4.5) and (3.5),

$\displaystyle\widetilde{\lambda}_{\infty}$
$\displaystyle(\mu,z)=\inf\left\\{\int|f|^{2}d\mu,~{}f\in
A_{e}(\Gamma),~{}f(z)=1\right\\}$
$\displaystyle=\inf\left\\{\int|f|^{2}d\Phi_{*}\mu,~{}f\in
A_{e}(\mathbb{T}),~{}f(\Phi(z))=1\right\\}=\lim_{n\to\infty}|\Phi(z)|^{2n}\lambda_{n}(\Phi_{*}\mu,\Phi(z))$
$\displaystyle=\lim_{n\to\infty}\lambda_{n}(\Phi_{*}\mu,1/\overline{\Phi(z)})=\lambda_{\infty}(\Phi_{*}\mu,1/\overline{\Phi(z)}).$

∎

We next prove a weak monotonicity of the sequence
$\\{\widetilde{B}_{n}(\mu,z)\\}_{n}$ for $\mu$ on $\Gamma$.

######  Lemma 4.4.

Let $z\in\Omega$ be fixed. Let $\mu$ be any measure supported on $\Gamma$ such
that the orthogonal polynomials are well-defined up to degree $N$. Let $n<N$.
Then there exist positive numbers $c_{n}\geq 1$ such that

$\widetilde{B}_{N-n}(\mu,z)\leq c_{n}\widetilde{B}_{N}(\mu,z),\qquad
c_{n}=1+\mathcal{O}\left(r^{n}\right),\qquad\text{as }n,N\to\infty,$ (4.8)

where the $c_{n}$’s are independent of the measure $\mu$.

###### Proof.

Let $n<N$. We will prove that

$|\Phi(z)|^{2n}\lambda_{N}(\mu,z)\leq c_{n}\lambda_{N-n}(\mu,z)$

for appropriate $c_{n}$, which is equivalent to the inequality in (4.8). Let
$R_{z}:=|\Phi(z)|>1$. We have

$\displaystyle|\Phi(z)|^{2n}\lambda_{N}(\mu,z)$
$\displaystyle=|\Phi(z)|^{2n}\min_{p\in\mathcal{P}_{N},~{}p(z)\neq
0}\frac{\int_{\Gamma}|p|^{2}d\mu}{|p(z)|^{2}}$
$\displaystyle\leq|\Phi(z)|^{2n}\min_{p\in\mathcal{P}_{N-n},~{}p(z)\neq
0}\frac{\int_{\Gamma}|F_{n}p|^{2}d\mu}{|F_{n}(z)p(z)|^{2}}$
$\displaystyle\leq\left(1+\mathcal{O}\left(\frac{1}{R_{z}^{n/2}}\right)\right)\min_{p\in\mathcal{P}_{N-n},~{}p(z)\neq
0}\frac{\int_{\Gamma}|F_{n}p|^{2}d\mu}{|p(z)|^{2}},$

where we have chosen $r=\sqrt{R_{z}}$ in (4.1). Next, for $\epsilon>0$ we fix
$r$ with $1-\epsilon<r<1$. For the numerator in the minimum,

$\displaystyle\int_{\Gamma}|F_{n}p|^{2}d\mu$
$\displaystyle\leq\|F_{n}\|^{2}_{\Gamma}\int_{\Gamma}|p|^{2}d\mu$
$\displaystyle\leq\left(1+\mathcal{O}\left(\frac{r^{n}}{(1-r)}\right)\right)\int_{\Gamma}|p|^{2}d\mu,$

where we have used $R=1$ and $r$ in (4.1). From the above, we obtain

$|\Phi(z)|^{2n}\lambda_{N}(\mu,z)\leq\left(1+\mathcal{O}\left(r^{n}\right)\right)\lambda_{N-n}(\mu,z),$

from which the existence of the $c_{n}$’s follows. The proof shows that they
are independent of the measure $\mu$. ∎

######  Remark 4.5.

In the particular case of $\Gamma=\mathbb{T}$, the unit circle, and
$\mu=d\theta/2\pi$, the family $\\{z^{n}\\}_{n\in\mathbb{N}}$ is an
orthonormal basis, and

$B_{n}(\mu,z)=\frac{|z|^{2n+2}-1}{|z|^{2}-1}.$

The inequality $\widetilde{B}_{n-1}(\mu,z)\leq\widetilde{B}_{n}(\mu,z)$ is
true since it is equivalent to $|z|^{2n+2}-1\leq|z|^{2n+2}$. For the harmonic
measure $\mu_{P}$ in (3.1), from (3.8) we have
$B_{n}(\mu_{P},z_{0})=|z_{0}|^{2n}$ so that
$\widetilde{B}_{n}(\mu_{P},z_{0})=1$ for all $n$.

In Proposition 4.6 and Lemma 4.7, the point $z$ is fixed and for any measure
$\mu$, we will simply write
$\widetilde{B}_{n}(\mu),\widetilde{B}_{\infty}(\mu)$ instead of
$\widetilde{B}_{n}(\mu,z),\widetilde{B}_{\infty}(\mu,z)$, and similarly for
other expressions depending on $z$.

######  Proposition 4.6.

Fix $z\in\Omega$. Assume that a subsequence $\\{\nu_{\varphi(n)}\\}_{n}$ of a
sequence $\\{\nu_{n}\\}_{n}$ of OPM’s tends weak-* to a limit measure
$\alpha$. Then $\alpha$ satisfies the following:
1) For all integers $k$, we have

$\widetilde{B}_{k}(\alpha)\leq 1\leq\widetilde{\lambda}_{k}(\alpha).$ (4.9)

2) $\alpha$ has an infinite number of points in its support.

###### Proof.

To show 1), for a given $k$,

$\displaystyle\widetilde{B}_{k}(\alpha)$
$\displaystyle=\left(\sup_{p\in\mathcal{P}_{k},~{}p(z)=1}\frac{1}{\int_{K}|p|^{2}d\alpha}\right)|\Phi(z)|^{-2k}$
$\displaystyle\leq\left(\liminf_{n}\sup_{p\in\mathcal{P}_{k},~{}p(z)=1}\frac{1}{\int_{K}|p|^{2}d\nu_{\varphi(n)}}\right)|\Phi(z)|^{-2k}$
(4.10)
$\displaystyle=\liminf_{n}\widetilde{B}_{k}(\nu_{\varphi(n)})\leq\liminf_{n}c_{\phi(n)-k}\widetilde{B}_{\phi(n)}(\nu_{\varphi(n)})=\liminf_{n}\widetilde{B}_{\phi(n)}(\nu_{\varphi(n)})\leq
1,$

where the first inequality on the last line and the final equality use Lemma
4.4, while the final inequality uses (1.3) and the fact that
$|\Phi(z)|=e^{g_{\Omega}(z)}$, $z\in\Omega$. The second inequality in (4.9) is
equivalent to the first one.
We prove 2) by contradiction. Assume that $\alpha$ has, say, $k$ points in its
support. Then, $B_{k}(\alpha)=\infty$, hence
$\widetilde{B}_{k}(\alpha)=\infty$, which contradicts the first inequality in
(4.9), and proves 2). Note, in particular, that orthogonal polynomials of all
degrees are well-defined for the measure $\alpha$. ∎

From (4.10), all numbers $\widetilde{B}_{n}(\alpha)$, $n\geq 0$, are less than
or equal to 1, and thus

$\widetilde{B}_{\infty}(\alpha)=\lim_{n}\widetilde{B}_{n}(\alpha)\leq 1$

(recall from Lemma 4.2 the limit exists).

######  Lemma 4.7.

Let $\\{\nu_{n}\\}_{n}$ be a sequence of OPM’s on $K$, with $\nu_{n}$ of order
$n$. For any subsequence $\\{\nu_{\varphi_{1}(n)}\\}_{n}$ of
$\\{\nu_{n}\\}_{n}$ with a weak-⋆ limit $\alpha$, there is a subsequence
$\\{\nu_{\varphi_{2}(n)}\\}_{n}$ of $\\{\nu_{\varphi_{1}(n)}\\}_{n}$ such that

$\lim_{n}\widetilde{B}_{\varphi_{2}(n)}(\nu_{\varphi_{2}(n)})=\widetilde{B}_{\infty}(\alpha).$
(4.11)

###### Proof.

Note that the weak-* convergence $\nu_{\varphi_{1}(n)}\to\alpha$ implies that
orthogonal polynomials for the limit measure $\alpha$ exist for any degree
$n\geq 0$ and

$\widetilde{B}_{\infty}(\alpha)=\lim_{n}\widetilde{B}_{\varphi_{1}(n)}(\alpha).$

We choose the subsequence $\\{\nu_{\varphi_{2}(n)}\\}_{n}$ of
$\\{\nu_{\varphi_{1}(n)}\\}_{n}$ in such a way that

$\forall n\geq 1,\quad\varphi_{2}(n-1)+n\leq\varphi_{2}(n).$ (4.12)

For a measure $\mu$, we set

$\widetilde{C}_{n}(\mu):=\left(\prod_{k=0}^{n}c_{k}\right)\widetilde{B}_{\varphi_{2}(n)}(\mu),\qquad
n\in\mathbb{N},$ (4.13)

where the $c_{k}$ are the constants in (4.8) (recall that they are independent
of $\mu$). From (4.8) and the condition (4.12), the sequence
$\widetilde{C}_{n}(\mu)$ is increasing with $n$. For the measure $\alpha$ we
also define

$\widetilde{C}_{\infty}(\alpha):=L\widetilde{B}_{\infty}(\alpha),\qquad
L:=\prod_{k=0}^{\infty}c_{k}\geq 1,$

The infinite product in the definition of $L$ converges because of the
asymptotic behavior of the $c_{k}$ as $k$ tends to infinity, see (4.8). Also,
by the choice of the subsequence $\\{\nu_{\varphi_{1}(n)}\\}_{n}$, we have

$\widetilde{C}_{\infty}(\alpha)=\lim_{n\to\infty}\widetilde{C}_{n}(\alpha).$
(4.14)

The set of measures
$S=\\{\nu_{\varphi_{2}(0)},\nu_{\varphi_{2}(1)},\ldots,\alpha\\}$ is compact.
Consider the array of values taken by the functions
$F_{0},F_{1},\ldots,F_{n},\ldots,F_{\infty}$ on $S$:

$F_{\infty}$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{\infty}(\alpha)$
---|---|---|---|---|---|---
$\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$
$F_{n}$ | $\widetilde{C}_{\infty}(\alpha)$ | $\ldots$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{n}(\nu_{\varphi_{2}(n)})$ | $\to$ | $\widetilde{C}_{n}(\alpha)$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
$F_{1}$ | $\widetilde{C}_{\infty}(\alpha)$ | $\widetilde{C}_{1}(\nu_{\varphi_{2}(1)})$ | $\ldots$ |  | $\to$ | $\widetilde{C}_{1}(\alpha)$
$F_{0}$ | $\widetilde{C}_{0}(\nu_{\varphi_{2}(0)})$ | $\widetilde{C}_{0}(\nu_{\varphi_{2}(1)})$ | … |  | $\to$ | $\widetilde{C}_{0}(\alpha)$
| $\nu_{\varphi_{2}(0)}$ | $\nu_{\varphi_{2}(1)}$ | $\ldots$ | $\nu_{\varphi_{2}(n)}$ | $\to$ | $\alpha$

where all values above the ascending main diagonal
$\widetilde{C}_{0}(\nu_{\varphi_{2}(0)}),\widetilde{C}_{1}(\nu_{\varphi_{2}(1)}),\ldots,\widetilde{C}_{n}(\nu_{\varphi_{2}(n)}),\ldots$
are equal to $\widetilde{C}_{\infty}(\alpha)$. The following properties are
satisfied:
a) The function $F_{\infty}$ is constant, hence continuous on $S$.
b) For each $n$, $F_{n}$ is continuous at $\alpha$ because
$\widetilde{C}_{n}(\nu_{\varphi_{2}(k)})\to\widetilde{C}_{n}(\alpha)$ as
$n\leq k\to\infty$. To see this, using (4.13) we have

$\widetilde{C}_{n}(\varphi_{2}(k))=\left(\prod_{k=0}^{n}c_{k}\right)\widetilde{B}_{\varphi_{2}(n)}(\nu_{\varphi_{2}(k)})$

and
$\widetilde{B}_{\varphi_{2}(n)}(\nu_{\varphi_{2}(k)})\to\widetilde{B}_{\varphi_{2}(n)}(\alpha)$
as $k\to\infty$ since $\nu_{\varphi_{2}(k)}\to\alpha$ (an orthogonal
polynomial of a given degree is a continuous function of the measure of
orthogonality; equivalently, the Bergman functions only depend on the moments
of a measure).
c) At each $\nu_{\varphi_{2}(n)}$, the sequence of functions
$F_{0},F_{1},\ldots,F_{n},\ldots$ increases to
$\widetilde{C}_{\infty}(\alpha)$. Indeed, by (4.8), we have

$\forall k\leq
n-1,~{}\widetilde{C}_{k}(\nu_{\varphi_{2}(n)})\leq\widetilde{C}_{k+1}(\nu_{\varphi_{2}(n)}),\qquad\text{and}\qquad\widetilde{C}_{n}(\nu_{\varphi_{2}(n)})\leq\widetilde{C}_{n}(\alpha)\leq\widetilde{C}_{\infty}(\alpha),$

where the next-to-last inequality uses that $\nu_{\varphi_{2}(n)}$ is an
optimal prediction measure.
d) At $\alpha$, the sequence of functions $F_{0},F_{1},\ldots,F_{n},\ldots$
also increases to $\widetilde{C}_{\infty}(\alpha)$. This is a consequence of
(4.8) and (4.14).

Hence, from Dini’s theorem, we may conclude that the convergence is uniform
which implies that
$\widetilde{C}_{n}(\nu_{\varphi_{2}(n)})\to\widetilde{C}_{\infty}(\alpha)$ and
thus also (4.11). ∎

###### Proof of Theorem 1.1.

We first assume that the curve $\Gamma$ is real analytic. Let
$\\{\nu_{\varphi_{1}(n)}\\}_{n}$ be a subsequence of $\\{\nu_{n}\\}_{n}$ which
converges weak-* to a probability measure $\alpha$. From Lemma 4.7, there
exists a subsequence $\\{\nu_{\varphi_{2}(n)}\\}_{n}$ of
$\\{\nu_{\varphi_{1}(n)}\\}_{n}$ such that, as $n$ tends to infinity,

$\widetilde{B}_{\varphi_{2}(n)}(\nu_{\varphi_{2}(n)},z_{0})\to\widetilde{B}_{\infty}(\alpha,z_{0}).$

By definition of the OPM’s,

$\forall\mu\in\mathcal{M}(\Gamma),~{}B_{\varphi_{2}(n)}(\nu_{\varphi_{2}(n)},z_{0})\leq
B_{\varphi_{2}(n)}(\mu,z_{0});\quad\hbox{hence}\
\widetilde{B}_{\varphi_{2}(n)}(\nu_{\varphi_{2}(n)},z_{0})\leq\widetilde{B}_{\varphi_{2}(n)}(\mu,z_{0}).$

Letting $n$ tend to infinity, we get
$\widetilde{B}_{\infty}(\alpha,z_{0})\leq\widetilde{B}_{\infty}(\mu,z_{0})$
which shows that $\alpha$ minimizes $\widetilde{B}_{\infty}(\mu,z_{0})$ over
$\mu\in\mathcal{M}(\Gamma)$, or equivalently, maximizes

$\widetilde{\lambda}_{\infty}(\mu,z_{0})=\liminf_{n\to\infty}\widetilde{\lambda}_{n}(\mu,z_{0})$

over measures $\mu\in\mathcal{M}(\Gamma)$. By Corollary 4.3, this is
equivalent to the fact that $\Phi_{*}\alpha$ maximizes
$\lambda_{\infty}(\nu,1/\overline{\Phi(z)})$ over measures
$\nu\in\mathcal{M}(\mathbb{T})$. Finally, Theorem 3.2 shows that

$\Phi_{*}\alpha=\widehat{\delta}_{1/\overline{\Phi(z_{0})}}=\widehat{\delta}_{\Phi(z_{0})},$

where the balayage is onto $\mathbb{T}$. By conformal invariance of the
balayage, we obtain that $\alpha$ equals $\widehat{\delta}_{z_{0}}$, the
balayage of $\delta_{z_{0}}$ onto $\Gamma$.

Using a classical estimate on Faber polynomials, see [10, Theorem 2 p.68], we
can now weaken the hypothesis on the regularity of $\Gamma$. Indeed, for
$\Gamma\in C(2,\alpha)$, $0<\alpha<1$ (i.e. $\Gamma$ can be parameterized by a
function of class $C(2,\alpha)$), the following estimate holds:

$F_{n}(z)=\Phi^{n}(z)\left(1+\mathcal{O}\left(\frac{\log
n}{n^{2+\alpha}}\right)\right),\quad n\to\infty,$

uniformly for $z\in\Gamma_{R}$, $R\geq 1$. It is easy to check that, making
use of that estimate in the proof of Lemma 4.4, leads to inequality (4.8)
where the estimate on $c_{n}$ is replaced with

$c_{n}=1+\mathcal{O}\left(\frac{\log n}{n^{2+\alpha}}\right),$

which is sufficient for the proof of Theorem 1.1, since we only need that the
infinite product of the $c_{n}^{\prime}s$ converges (see the proof of Lemma
4.7). ∎

We end with a discussion of a related asymptotic problem. For a connected,
simply connected, compact subset $K$ of $\mathbb{C}$ we recall from (1.3) that
for $z_{0}\in\Omega$,

$B_{n}(\nu_{n},z_{0})=\sup_{p\in\mathcal{P}_{n}}\frac{|p(z_{0})|^{2}}{\|p\|_{K}^{2}}\leq
e^{2ng_{\Omega}(z_{0})}=|\Phi(z_{0})|^{2n}.$

In fact, from the first equality together with (1.4) it follows that

$\lim_{n\to\infty}\frac{B_{n}(\nu_{n},z_{0})^{1/2n}}{|\Phi(z_{0})|}=\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})^{1/2n}=1.$

There is the deeper question as to whether the limit of the sequence
$\\{\widetilde{B}_{n}(\nu_{n},z_{0})\\}_{n}$ – without the $1/2n$ power –
exists. Clearly

$0\leq\liminf_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})\leq\limsup_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})\leq
1.$

  1. 1.

For the case of the unit circle, since

$B_{n}(\nu_{n},z_{0})=\sup_{p\in\mathcal{P}_{n}}\frac{|p(z_{0})|^{2}}{\|p\|_{K}^{2}}=|z_{0}|^{2n}=|\Phi(z_{0})|^{2n},$

recall (3.1) and (3.8), we have $\widetilde{B}_{n}(\nu_{n},z_{0})=1$ for all
$n$.

  2. 2.

As a corollary of Lemma 4.7 and Theorem 1.1, it follows that for a real
analytic curve $\Gamma$, we have

$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})=1$ (4.15)

for all $z_{0}\in\Omega$.

  3. 3.

For the interval $[-1,1]$, the existence of this limit for
$z_{0}\not\in[-1,1]$ was shown by Yuditskii [12] and Peherstorfer [6]; their
proofs are very technical. Writing $\psi(z):=z+\sqrt{z^{2}-1}$ for the
conformal map from $\mathbb{C}\setminus[-1,1]$ onto
$\mathbb{C}\setminus\mathbb{D}$, we have $g_{\Omega}(z)=\log|\psi(z)|$. Two
special cases are more easily computed. First, for
$x\in\mathbb{R}\setminus[-1,1]$, the polynomial $p_{n}$ in (2.2) is the
Chebyshev polynomial

$T_{n}(z)={1\over 2}\left((\psi(z))^{n}+({\psi(z)})^{-n}\right).$

Thus for such $x$, from (2.1),

$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},x)=\lim_{n\to\infty}\frac{1}{2}\frac{|(\psi(x))^{n}+({\psi(x)})^{-n}|}{|\psi(x)|^{n}}=\frac{1}{2}.$

Next, for $z=ia,\ a\in\mathbb{R},\ |a|>1$, from [3]

$|p_{n}(ia)|=\sqrt{a^{2}+1}[|a|+\sqrt{a^{2}+1}]^{n-1}.$

Since $|\psi(ia)|=|a+\sqrt{a^{2}+1}|$, we have, for $a>0$,

$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},ia)=\lim_{n\to\infty}\frac{\sqrt{a^{2}+1}[|a|+\sqrt{a^{2}+1}]^{n-1}}{|a+\sqrt{a^{2}+1}|^{n}}=\frac{\sqrt{a^{2}+1}}{a+\sqrt{a^{2}+1}}.$

The results in [12] and [6] seem to indicate that, as with these special
cases, for any $z_{0}\not\in[-1,1]$,

$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})<1.$ (4.16)

  4. 4.

For a circular arc $A_{\alpha}:=\\{z\in\mathbb{C}:|z|=1,\
|\hbox{arg}z|\leq\alpha\\},\ 0<\alpha<\pi$, Eichinger [4] shows that
$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})$ exists for any $z_{0}$
with $|z_{0}|\not=1$ and he calculates this limit.

Concerning 2., in particular, for the confocal ellipses

$E_{r}:=\\{z\in\mathbb{C}:|z-1|+|z+1|=r+1/r\\}$

(4.15) holds for all points $z_{0}$ outside $E_{r}$ for each $r>1$. As $r\to
1$, these ellipses converge to the interval $[-1,1]$, which, according to 3.,
fails to have this property. We know of no general results on existence of the
limit of the sequence $\\{\widetilde{B}_{n}(\nu_{n},z_{0})\\}_{n}$.

######  Remark 4.8.

For the interval $[-1,1]$, or, more generally, for a real analytic arc
$\gamma$, the analogues of Lemma 4.2 and Corollary 4.3 hold, using the
appropriate generalization of the space $A_{e}(\Gamma)$. However, there is a
problem with generalizing the “weak monotonicity” lemma, Lemma 4.4. Indeed, if
such a result were true for $[-1,1]$, then the proofs of Proposition 4.6 and
hence Lemma 4.7 and Theorem 1.1 would be valid as well. However, equation
(4.15) then gives

$\lim_{n\to\infty}\widetilde{B}_{n}(\nu_{n},z_{0})=1$

which contradicts (4.16). Thus other ideas or techniques are required to deal
with arcs.

## References

  * [1] N. I. Achiezer, Theory of approximation, F. Ungar, 1956.
  * [2] A. Ambroladze, H. Wallin, Convergence of rational interpolants with preassigned poles. J. Approx. Theory 89 (1997), no. 2, 238-256.
  * [3] L. Bos, N. Levenberg, J. Ortega-Cerdà, Optimal polynomial prediction measures and extremal polynomial growth. Constr. Approx. 54 (2021), 431-453.
  * [4] B. Eichinger, Szegő-Widom asymptotics of Chebyshev polynomials on circular arcs. J. Approx. Theory 217 (2017), 15-25.
  * [5] G. Meinardus, Approximation of functions: Theory and numerical methods. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag, New York 1967.
  * [6] F. Peherstorfer, Extremal problems of Chebyshev type. Proc. Amer. Math. Soc. 137 (2009), no. 7, 2351-2361.
  * [7] T. Ransford, _Potential theory in the complex plane_ , Cambridge University Press, Cambridge, 1995.
  * [8] E. B. Saff and V. Totik,  Logarithmic Potentials with External Fields, volume 316 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997.
  * [9] B. Simon, Orthogonal polynomials on the unit circle. Part 1. Classical theory. Amer. Math. Soc. Colloquium Publications 54, Part 1. American Mathematical Society, Providence, RI, 2005.
  * [10] P.K. Suetin, Series of Faber polynomials. Analytical Methods and Special Functions, Gordon and Breach Science Publishers, Amsterdam, 1998.
  * [11] W. Van Assche, Orthogonal polynomials in the complex plane and on the real line. Special functions, q-series and related topics (Toronto, ON, 1995), 211-245, Fields Inst. Commun. 14, Amer. Math. Soc., Providence, RI, 1997.
  * [12] P. Yuditskii, A complex extremal problem of Chebyshev type. J. Anal. Math. 77 (1999), 207-235.

S. Charpentier<EMAIL_ADDRESS>
F. Wielonsky<EMAIL_ADDRESS>
Laboratoire I2M - UMR CNRS 7373
Université Aix-Marseille, CMI 39 Rue Joliot Curie
F-13453 Marseille Cedex 20, FRANCE

N. Levenberg<EMAIL_ADDRESS>
Indiana University, Bloomington, IN 47405 USA