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Weighted polynomial approximation in the complex plane. (English) Zbl 0963.30024
Summary: Given a pair \((G,W)\) of an open bounded set \(G\) in the complex plane and a weight function \(W(z)\) which is analytic and different from zero in \(G\), we consider the problem of the locally uniform approximation of any function \(f(z)\), which is analytic in \(G\), by weighted polynomials of the form \(\left \{W^{n}(z)P_{n}(z) \right \}^{\infty}_{n=0}\), where \(\deg P_{n} \leq n\). The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.
MSC:
30E10 Approximation in the complex plane
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
41A30 Approximation by other special function classes
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