Weighted holomorphic polynomial approximation (Q6583666)

The paper is concerned with the weighted holomorphic polynomial approximation. In the late 1990's \textit{I. E. Pritsker} and \textit{R. S. Varga} [Constr. Approx. 14, No. 4, 475--492 (1998; Zbl 0920.30028)] showed that any function holomorphic in an open set \(G\) in \(\mathbb{C}\) can be locally uniformly approximated in \(G\) by weighted holomorphic polynomials \(\{W(z)^{n}p_{n}(z)\}, \deg (p_{n})\le n\), where \(W\) is a non-vanishing holomorphic function in \(G\). In this paper this theory has been developed by proving a quantitative Bernstein-Walsh type theorem for certain pairs \((G,W)\). A special case has been considered where \(W(z)=\frac{1}{(z+1)}\) and \(G\) is a loop of the lemniscate \(\{z\in\mathbb{C}:|z(z+1)|=\frac{1}{4}\}\). It has been shown that the normalized measures associated to the zeros of the \(n^{th}\) order Taylor polynomial about \(0\) of the function \((1+z)^{-n}\) converge to the weighted equilibrium measure of \({G}\) with weight \(|W|\) as \(n\rightarrow\infty\). In the last weighted holomorphic polynomial approximation has been studied in \(\mathbb{C}^{n}\), \(n>1\).