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

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For a compact set E E with connected complement, let A ( E ) A(E) be the uniform algebra of functions continuous on E E and analytic interior to E . E. We describe A ( E , W ) , A(E,W), the set of uniform limits on E E of sequences of the weighted polynomials { W n ( z ) P n ( z ) } n = 0 ∞ , \{W^n(z)P_n(z)\}_{n=0}^{\infty }, as n → ∞ , n \to \infty , where W ∈ A ( E ) W \in A(E) is a nonvanishing weight on E . E. If E E has empty interior, then A ( E , W ) A(E,W) is completely characterized by a zero set Z W ⊂ E . Z_W \subset E. However, if E E is a closure of Jordan domain, the description of A ( E , W ) 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 ) . A(E,W).