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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Polynomial approximation with varying weights on compact sets of the complex plane
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by Igor E. Pritsker PDF
Proc. Amer. Math. Soc. 126 (1998), 3283-3292 Request permission


For a compact set $E$ with connected complement, let $A(E)$ be the uniform algebra of functions continuous on $E$ and analytic interior to $E.$ We describe $A(E,W),$ the set of uniform limits on $E$ of sequences of the weighted polynomials $\{W^n(z)P_n(z)\}_{n=0}^{\infty },$ as $n \to \infty ,$ where $W \in A(E)$ is a nonvanishing weight on $E.$ If $E$ has empty interior, then $A(E,W)$ is completely characterized by a zero set $Z_W \subset E.$ However, if $E$ is a closure of Jordan domain, the description of $A(E,W)$ also involves an inner function. In both cases, we exhibit the role of the support of a certain extremal measure, which is the solution of a weighted logarithmic energy problem, played in the descriptions of $A(E,W).$
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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
  • Address at time of publication: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7058
  • MR Author ID: 319712
  • Email:,
  • Received by editor(s): September 4, 1996
  • Received by editor(s) in revised form: March 25, 1997
  • Communicated by: Theodore W. Gamelin
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3283-3292
  • MSC (1991): Primary 30E10; Secondary 30B60, 31A15, 41A30
  • DOI:
  • MathSciNet review: 1452821