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\begin{document}

%\markboth{Igor E. Pritsker and Richards S. Varga}{Weighted Polynomial
%Approximation in the Complex Plane}

\title{
Weighted polynomial approximation in the complex plane}

% author one information
\author{Igor E. Pritsker}
\address{Institute for Computational Mathematics, %\\
Department of Mathematics and Computer Science, %\\
Kent State University, Kent, Ohio 44242-0001}
\email{pritsker@mcs.kent.edu}
%
% author two information
\author{Richard S. Varga}
\address{Institute for Computational Mathematics, %\\
Department of Mathematics and Computer Science, %\\
Kent State University, Kent, Ohio 44242-0001}
\email{varga@mcs.kent.edu}
%\thanks{}

\subjclass{Primary 30E10; Secondary 30C15, 31A15, 41A30}

\date{October 15, 1996}

\keywords{Weighted polynomials, locally uniform approximation, 
logarithmic potential, balayage}

\copyrightinfo{1997}{American Mathematical Society}

\commby{Yitzhak Katznelson}

%\dedicatory{}

\begin{abstract}
Given a pair $(G,W)$ of an open bounded set $G$ in the complex plane
and a weight function $W(z)$ which is analytic and different from
zero in $G$, we consider the problem of the locally uniform
approximation of any function $f(z)$, which is analytic in $G$, by
weighted polynomials of the form $\left\{W^{n}(z)P_{n}(z)
\right\}^{\infty}_{n=0}$, where $\deg P_{n} \leq n$.  The main
result of this paper is a necessary and sufficient condition for such
an approximation to be valid.  We also consider a number of applications
of this result to various classical weights, which give explicit
criteria for these weighted approximations.
\end{abstract}

\maketitle


\section{Introduction and general result} \label{sec1}
In this paper, we will examine pairs of the form

\begin{equation} \label{1.1a}
(G,W)
\end{equation}
where 
\begin{equation} \label{1.1}
\ \ \ \left\{ \begin{array}{lp{3.8in}}
\mathrm{(i)} & $G$ is an open bounded set, in the complex plane 
${\C}$, which can be represented as a finite or countable union of
disjoint simply connected domains, i.e., $G=\bigcup^{\sigma}_{\ell=1}
G_{\ell}$ (where $1 \leq \sigma \leq \infty$ );\\
\mathrm{(ii)} & $W(z)$, the weight function, is analytic in  $G$ with 
$W(z) \neq 0$ for any  $z \in G$.
\end{array} \right.
\end{equation}
We say that the pair ($G,W$) has the {\bf approximation property} if,
\begin{equation} \label{1.1b}
\ \ \ \left\{ \begin{array}{p{4in}}
for any $f(z)$ 
which is analytic in $G$ and for any compact subset 
$E$ of $G$, there exists a sequence of polynomials 
$\{P_{n}(z)\}^{\infty}_{n=0}$, with $\deg P_{n} \leq n$
for all $n \geq 0$, such that \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ $\dis\lim_{n \rightarrow
\infty} \|f-W^{n} P_{n} \|_{E} =0$, 
\end{array} \right.
\end{equation}
where all norms throughout this paper are the uniform (Chebyshev)
norms on the indicated sets.

Given a pair $(G,W)$, as in (\ref{1.1}), we state our 
main result, Theorem \ref{th1.1}, which gives a characterization,
in terms of potential theory, for the pair $(G,W)$ to have the
approximation property.
For notation, let ${\mathcal{M}}(E)$ be the space of all positive unit Borel
measures on ${\C}$ which are supported on a compact set $E$,
i.e., for any $\mu \in {\mathcal{M}}
(E)$, we have $\mu ({\C})=1$ and supp~$\!\mu \subset 
E$.  The logarithmic potential of a compactly supported
measure $\mu$ is defined (cf. Tsuji \cite[p. 53]{T}) by
\begin{equation} \label{1.2}
U^{\mu}(z):= \int \log \frac{1}{|z-t|} \ d \mu (t).
\end{equation}

\begin{theorem} \label{th1.1}
A pair $(G,W)$, as in {\rm (\ref{1.1})}, has the  approximation property
{\rm (\ref{1.1b})} if and only
if there exist a measure $\mu (G,W) \in {\mathcal{M}} (\partial G)$ 
and a constant $F(G,W)$ such that 
\begin{equation} \label{1.3}
U^{\mu(G,W)}(z)- \log |W(z)|=F(G,W), \quad \mbox { for any } z \in G.
\end{equation}
\end{theorem}


\begin{remark} \label{re1.2}
{\rm It is well known that any open set in the complex plane is a finite
or countable union of disjoint domains, and this is more general than the
assumption on the open set $G$ in \eqref{1.1}(i).  However, we note that
the approximation property (\ref{1.1b}) {\em cannot hold}, 
even in the classical case where $W(z) \equiv 1$ for all $z \in G$,
if $G = \bigcup^{\sigma}_{\ell = 1} G_{\ell}$,
when some $G_{\ell}$ is multiply connected  (cf. Walsh \cite[p. 25]{W}). 
In this sense, our initial assumptions on $G$ are quite general.}
\end{remark}

\begin{remark} \label{re1.3}
{\rm The condition that $W(z) \neq 0$ for all $z \in G$ cannot be dropped,
for if $W(z_{0})=0$ for some $z_{0} \in G_{\ell}$, where $G=
\bigcup^{\sigma}_{\ell = 1} G_{\ell}$, then the necessarily null sequence 
$\left\{ W^{n}(z_{0})P_{n}(z_{0}) \right\}^{\infty}_{n=0}$ trivially fails
to converge to any $f(z)$, analytic in $G$, with  
$f(z_{0}) \neq 0$; whence, the approximation property fails.  Even more
decisive is the result that if $W(z_{0})=0$
for some $z_{0} \in G_{\ell}$,
then the sequence $\left\{ W^{n}(z)P_{n}(z) \right\}^{\infty}_{n=0}$ can
converge, locally  uniformly in $G$, to $f(z)$, only if $f(z) \equiv 0$
in $G_{\ell}$.  In this sense, the assumptions
on $W(z)$ are also quite general.}
\end{remark}

\begin{remark} \label{re1.5}
{\rm In the case $W(z) \equiv 1$ of Theorem \ref{th1.1}, the result
that the approximation property (\ref{1.1b}) holds is a known classical
result in complex approximation theory (cf. \cite[p. 26]{W}).  This
also follows from Theorem \ref{th1.1} because the measure $\mu(G,1)$
exists by Theorems III.12 and III.14 of Tsuji \cite{T}, and is the
classical equilibrium distribution measure (in the sense of logarithmic
potential theory) for $\oG$.}
\end{remark}

The topic of weighted approximation by $\left\{W^{n}(z)P_{n}(z)
\right\}^{\infty}_{n=0}$, on the
real line, has been extensively and thoroughly treated in the recent
books of Saff and Totik \cite{ST} and Totik \cite{To}.  Here, we
emphasize weighted approximation {\em in the complex plane}, 
which has received far less attention in the current approximation
theory literature, with the exception of the recent papers by
Borwein and Chen \cite{BC} and Pritsker and Varga \cite{PV}.

We shall present in Section \ref{sec2} a number of applications
of Theorem \ref{th1.1} to special pairs $(G,W)$. The concluding 
Section \ref{sec5}
is devoted to 
further remarks, open problems and
a discussion of possible generalizations.

%\setcounter{equation}{0}
\section{Applications} \label{sec2}
Finding the measure $\mu(G,W)$ of Theorem \ref{th1.1} or verifying its
existence is a nontrivial problem in general.  Since 
$U^{\mu (G,W)}(z)$ is harmonic in ${\C} \backslash$  {\rm supp}~$\!\mu
(G,W)$ and since it can be derived from (\ref{1.3}) that if $\log$ $|W(z)|$
is continuous on $\oG$ and $G$ is a {\em finite} union
of $G_{\ell},\; \ell = 1, 2,\dots,\ell_{0}$, then
$U^{\mu(G,W)}(z)$ is
equal to $\log |W(z)|+F(G,W)$ on supp~$\!\mu(G,W)$, 
it follows that $U^{\mu(G,W)}(z)$
can be found as the solution of the corresponding Dirichlet problems. 
The measure $\mu (G,W)$ can be recovered from its
potential, using the Fourier method described in Section IV.2
of Saff and Totik \cite{ST}. 
%We refer the reader to Section \ref{sec3} of this paper 
%and to Chapter IV of \cite{ST} for details.
This method has already been used successfully 
by the authors in \cite{PV} to study the
approximation of analytic functions by the weighted polynomials
$\left\{ e^{-nz}P_{n}(z)\right\}^{\infty}_{n=0}$, i.e., when $W(z):=e^{-z},$
and it is also used in the proof of Theorem \ref{th2.6}.

In contrast to the above procedure, we next consider a different
method, dealing with specific weight functions, which allows us
to deduce ``explicit'' expressions for the measure $\mu(G,W)$ of
Theorem \ref{th1.1}, and to treat some
important cases of pairs $(G,W)$.
With $G$ as defined in \eqref{1.1}(i) for $\sigma$ {\it finite}, we
denote the unbounded component of $\overline{\C}\backslash\oG$ by $\Omega$.
Let $\nu_{1}$ and $\nu_{2}$ be two unit positive Borel
measures on ${\C}$ with compact supports satisfying
\begin{equation} \label{2.1}
\mbox {supp}~\!\nu_{1} \subset \overline{\C} \backslash G \mbox {\quad 
and\quad} 
\mbox {supp}~\!\nu_{2} \subset \overline{\C} \backslash G,
\end{equation}
such that
\begin{equation} \label{2.2}
\nu_{1}({\C}) = \nu_{2} ({\C})=1.
\end{equation}
For real numbers $\alpha$ and $\beta$, assume that $W(z)$, satisfying
\begin{equation} \label{2.3}
\log |W(z)|= - \left( \alpha U^{\nu_{1}}(z) + \beta U^{\nu_{2}}(z) \right),
\quad z \in G,
\end{equation}
is analytic in $G$.  Then, as an application of
Theorem \ref{th1.1}, we state our next result: 

\begin{theorem} \label{th2.1}
Given any pair of real numbers $\alpha$ and $\beta$, given an open bounded 
set $G= \bigcup^{\sigma}_{\ell=1} G_{\ell}$ as in {\rm \eqref{1.1}(i)} with
$\sigma$ finite, 
%$\left\{
%G_{\ell} \right\}^{\ell_{0}}_{\ell=1}$ is 
%a finite collection of disjoint simply connected domains,
and given the weight function $W(z)$ of {\rm(\ref{2.3})},
then the pair $(G,W)$
has the approximation property {\rm(\ref{1.1b})} if and only if the measure
\begin{equation} \label{2.4}
\mu:= (1 + \alpha + \beta) \omega (\infty, \cdot, \Omega)-
\alpha \hat{\nu}_{1}-\beta \hat{\nu}_{2}
\end{equation}
is positive, where $\omega (\infty, \cdot, \Omega)$ is the harmonic
measure at $\infty$ with respect to $\Omega$; here, $\hat{\nu}_{1}$ and
$\hat{\nu}_{2}$ are, respectively,  the balayages of
$\nu_{1}$ and $\nu_{2}$ from $\overline{\C} \backslash \oG$ to $\oG$.

Furthermore, if $\mu$ of {\rm (\ref{2.4})} is a positive measure, then
{\rm (cf. Theorem \ref{th1.1})}
\begin{equation} \label{2.5}
\mu (G,W) = \mu \ \mbox { and \ }
\mbox {\rm  supp}~\!\mu (G,W) \subset \partial G.
\end{equation}
\end{theorem}

We point out that the harmonic measure $\omega(\infty, \cdot, \Omega)$
(cf. Nevanlinna \cite{N} and Tsuji \cite{T}) is the same as the
equilibrium distribution
measure for $\oG,$ in the sense of classical logarithmic potential theory
\cite{T}.  For the notion of balayage of a measure, we refer the reader
to Chapter IV of Landkof \cite{La} or Section II.4 of Saff and
Totik \cite{ST}.

In the following series of subsections, we consider various classical
weight functions and find their corresponding measures, associated with
the weighted approximation problem in $G$ by Theorem \ref{th1.1}.

\subsection{Incomplete polynomials and Laurent polynomials} \label{sec2.1}
With ${\N}_{0}$ and ${\N}$ denoting respectively the sets of
nonnegative and positive integers, the 
{\em incomplete polynomials} of Lorentz \cite{Lo} are a
sequence of polynomials of the form
\begin{equation} \label{2.7}
\left\{ z^{m(i)} P_{n(i)} (z) \right\}^{\infty}_{i=0}, \quad \deg
P_{n(i)} \leq n (i), \ (m(i), n(i) \in {\N}_{0}),
\end{equation}
where it is assumed that $\dis\lim_{i \rightarrow \infty} \
\frac{m(i)}{n(i)}=:
\alpha$, where $\alpha > 0$ is a real number. 
The question of the possibility of approximation by incomplete
polynomials is closely connected to that of approximation by
the weighted polynomials 
\begin{equation} \label{2.8}
\left\{ z^{\alpha n} P_{n}(z) \right\}^{\infty}_{n=0}, \quad
\deg P_{n} \leq n.
\end{equation}
The question of approximation by the incomplete polynomials of
(\ref{2.7}) was completely settled by Saff and Varga \cite{SV}, and
by von Golitschek \cite{Go} on the interval $[0,1]$ (see Lorentz, 
von Golitschek, and Makovoz \cite{LGM}, Totik \cite{To}, and
Saff and Totik \cite{ST} for the associated history and later developments).
We consider now the analogous problem in the complex plane.  Since the
weight $W(z) := z^{\alpha}$ in (\ref{2.8}) is multiple-valued in
${\C}$ if $\alpha \notin {\N}_{0}$, 
we then restrict ourselves to the slit domain $S_{1}:=
{\C} \backslash (-\infty, 0]$ and the single-valued branch of
$W(z)$ in $S_{1}$ satisfying $W(1)=1$.

For the related question of the approximation by the so-called Laurent
polynomials
\begin{equation} \label{2.9}
\left\{ \dis\frac{P_{n(i)}(z)}{z^{m(i)}} \right\}^{\infty}_{i=0},\qquad
\deg P_{n(i)} \leq n(i) \quad (m(i), n(i) \in {\N}_{0}),  
\end{equation}
where $\dis\lim_{i \rightarrow \infty} \ \dis\frac{m(i)}{n(i)} :=
\alpha, \alpha > 0$, we are similarly led to the question of the
approximation by the weighted polynomials  
\begin{equation} \label{2.10}
\left\{ z^{-\alpha n}P_{n}(z) \right\}^{\infty}_{n=0},\qquad \deg
P_{n} \leq n,
\end{equation}
with the only difference being in the sign in the exponent of the weight
function.  Thus, we can give a unified treatment of both problems by
considering weighted approximation by $\left\{ W^{n}(z) P_{n}(z)
\right\}^{\infty}_{n=0}, \deg P_{n} \leq n$, with 
\begin{equation} \label{2.11}
W(z):= z^{\alpha}, \quad z \in S_{1}:= {\C} \backslash (-\infty,0],
\end{equation}
where $\alpha$ is {\em any} fixed real number and 
where we choose, as before, the single-valued branch of $W(z)$ in
$S_{1}$ satisfying $W(1)=1$.

\begin{theorem} \label{th2.2}
Given an open set $G$ as in {\rm \eqref{1.1}(i)} with $\sigma$ finite,
such that $\oG \subset
S_{1}$, and given the weight function
$W(z)$ of {\rm (\ref{2.11})}, then the pair $(G,W)$ has the
approximation property {\rm (\ref{1.1b})} if and only if
\begin{equation} \label{2.12}
\mu = (1 + \alpha) \omega(\infty, \cdot, \Omega)-\alpha \omega
(0, \cdot, \Omega)
\end{equation}
is a positive measure, where $\omega (\infty, \cdot, \Omega)$ and
$\omega (0, \cdot, \Omega)$ are, respectively, the harmonic measures
with respect to the unbounded component $\Omega$ of\/ $\overline{\C}
\backslash \oG$, at $z= \infty$ and at $z=0$.
\end{theorem}

In some cases, when the geometric shape of $G$ is given explicitly,
we can determine the explicit form of the measure of {\rm (\ref{2.12})}.  This
is especially easy to do for disks.

\begin{corollary} \label{coro2.3}
Given the disk $D_{r}(a):= \left\{ z \in {\C}:|z-a|