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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.

Copyright 1997 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 03 (1997), pp. 38-44
• Publisher Identifier: S 1079-6762(97)00021-8
• 1991 Mathematics Subject Classification. Primary 30E10; Secondary 30C15, 31A15, 41A30
• Key words and phrases. Weighted polynomials, locally uniform approximation, logarithmic potential, balayage
• Received by the editors October 15, 1996
• Posted on May 2, 1997
• Communicated by Yitzhak Katznelson
• Comments (When Available)

Igor E. Pritsker

Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242-0001

E-mail address: pritsker@mcs.kent.edu

Richard S. Varga

Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242-0001

E-mail address: varga@mcs.kent.edu