Uniform Approximation of Extremal Functions in Weighted Bergman Spaces

Abstract

We discuss approximation of extremal functions by polynomials in the weighted Bergman spaces $A^p_\alpha$ where $-1 < \alpha < 0$ and $-1 < \alpha < p-2$. We obtain bounds on how close the approximation is to the true extremal function in the $A^p_\alpha$ and uniform norms. We also discuss several results on the relation between the Bergman modulus of continuity of a function and how quickly its best polynomial approximants converge to it.