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Polynomial approximation with varying weights on compact sets of the complex plane

Author: Igor E. Pritsker
Journal: Proc. Amer. Math. Soc. 126 (1998), 3283-3292
MSC (1991): Primary 30E10; Secondary 30B60, 31A15, 41A30
MathSciNet review: 1452821
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Abstract: 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

Keywords: Weighted polynomials, closed ideals, weighted energy problem, logarithmic potentials, uniform algebras
Received by editor(s): September 4, 1996
Received by editor(s) in revised form: March 25, 1997
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society