Weighted holomorphic polynomial approximation

For G an open set in \({\mathbb {C}}\) $\mathbb{C}$ and W a non-vanishing holomorphic function in G, in the late 1990’s, Pritsker and Varga (Constr Approx 14, 475-492 1998) characterized pairs (G, W) having the property that any f holomorphic in G can be locally uniformly approximated in G by weighted holomorphic polynomials \(\{W(z)^np_n(z)\}, \ deg(p_n)\le n\) $\{W{(z)}^n{p}_n(z)\},deg(p_n)\le n$ . 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 \(W(z)=1/(1+z)\) $W(z)=1/(1+z)$ and G is a loop of the lemniscate \(\{z\in {\mathbb {C}}: |z(z+1)|=1/4\}\) $\{z\in \mathbb{C}:|z(z+1)|=1/4\}$ . We show the normalized measures associated to the zeros of the \(n-th\) $n-th$ order Taylor polynomial about 0 of the function \((1+z)^{-n}\) ${(1+z)}^{-n}$ converge to the weighted equilibrium measure of \({\overline{G}}\) $\overline{G}$ with weight |W| as \(n\rightarrow \infty \) $n\to \infty$ . This mimics the motivational case of Pritsker and Varga (Trans Amer Math Soc 349, 4085-4105 1997) where G is the inside of the Szegő curve and \(W(z)=e^{-z}\) $W(z)=e^{-z}$ . Lastly, we initiate a study of weighted holomorphic polynomial approximation in \({\mathbb {C}}^n, \ n>1\) ${\mathbb{C}}^n,n>1$ .