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Weighted polynomial approximation in the complex plane

Authors: Igor E. Pritsker and Richard S. Varga
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 38-44
MSC (1991): Primary 30E10; Secondary 30C15, 31A15, 41A30
Published electronically: May 2, 1997
MathSciNet review: 1445633
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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

Igor E. Pritsker
Affiliation: Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242-0001

Richard S. Varga
Affiliation: Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242-0001

Keywords: Weighted polynomials, locally uniform approximation, logarithmic potential, balayage
Received by editor(s): October 15, 1996
Published electronically: May 2, 1997
Communicated by: Yitzhak Katznelson
Article copyright: © Copyright 1997 American Mathematical Society

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