Weighted holomorphic polynomial approximation

Abstract

For $G$ an open set in $\mathbb{C}$ and $W$ a non-vanishing holomorphic function in $G$, in the late 1990's, Pritsker and Varga characterized pairs $(G,W)$ having the property that any $f$ holomorphic in $G$ can be locally uniformly approximated in $G$ by weighted holomorphic polynomials $\{W(z)^np_n(z)\}, \ deg(p_n)\leq n$. We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs $(G,W)$. Then we consider the special case where $W(z)=1/(1+z)$ and $G$ is a loop of the lemniscate $\{z\in \mathbb{C}: |z(z+1)|=1/4\}$. We show 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 $\overline G$ with weight $|W|$ as $n\to \infty$. This mimics the motivational case of Pritsker and Varga where $G$ is the inside of the Szego curve and $W(z)=e^{-z}$. Lastly, we initiate a study of weighted holomorphic polynomial approximation in $\mathbb{C}^n, \ n>1$.