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## Weighted polynomial approximation in the complex plane

### Igor E. Pritsker and Richard S. Varga

**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.

*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`

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