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Morgoulis \controldates{21-APR-1997,21-APR-1997,21-APR-1997,21-APR-1997} \documentclass{era-l} \issueinfo{3}{5}{January}{1997} \dateposted{May 2, 1997} \pagespan{38}{44} \PII{S 1079-6762(97)00021-8} \newcommand{\R}{\Bbb R} %\rm I\kern-.19emR} \newcommand{\M}{\Bbb M} %\rm I\kern-.19emM} \newcommand{\N}{\Bbb N} %\rm I\kern-.19emN} \newcommand{\C}{\Bbb C} %\rm I\kern-.5emC} \newcommand{\boldxi}{\mbox{\boldmath $\xi$} } \newcommand{\oG}{\overline{G}} \newcommand{\OG}{(\overline{G})} \newcommand{\dis}{\displaystyle} \newcommand{\Notequiv}{/\kern-.6em\hbox{$\equiv$} } \newcommand{\db}{\raisebox{-.75ex} {\rule[.05in]{.09in}{.10in} }}%darkbox \newcommand{\eop}{\hfill{\vrule height 7pt width 7pt depth 0pt}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{problem}{Problem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \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|