Weighted polynomial approximation in the complex plane
The paper is concerned with pairs \((G,W)\), where \(G\) is a bounded open set in the complex plane which can be represented as a finite or countable union of disjoint simply connected regions, and \(W(z)\), the weight function, is analytic in \(G\) with \(W(z)\neq 0\). The pair \((G,W)\) has the approximation property if any analytic function \(f\) in \(G\) can be represented on any compact \(E\subset G\) as the uniform limit of weighted polynomials \(W^nP_n\) as \(n\rightarrow \infty\), with polynomials \(P_n\) of degree \(\leq n\). The authors' main result is the criterion: The pair \((G,W)\) has the approximation property if and only if the harmonic function \(\log | W(z)| \) can be represented (up to an additive constant) as the logarithmic potential of a unit measure \(\mu\) with support in \(\partial G\). This result is applied to various classical weights and gives explicit criteria for these approximation problems. The measure \(\mu\) is related to the solution of a minimal weighted energy problem.
scientific article; zbMATH DE number 1268075