On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

We prove that a holomorphic function on a neighborhood of a compact convex set K⊂Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \subset {{\,\mathrm{{\mathbb {C}}}\,}}^n$$\end{document} can be uniformly on K approximated by polynomials with an error that decreases exponentially fast with the growth of the polynomial degree. The presented method is based on the vanishing of the top Dolbeault cohomology group of an open subset in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{{\mathbb {C}}}\,}}^n$$\end{document} and an argument involving Čech cohomology. In comparison with the Bernstein-Walsh approach previously applied to the problems of this type the method presented here is much more elementary but it does not provide effective estimates.