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

Pritsker, Igor

doi:10.1090/S0002-9939-98-04402-5

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

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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