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

DOI:
https://doi.org/10.1090/S0002-9939-98-04402-5

MathSciNet review:
1452821

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
319712

Email:
pritsker@mcs.kent.edu, iep@po.cwru.edu

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