# Uniform Approximation of Extremal Functions in Weighted Bergman Spaces

## TL;DR

This paper studies how well extremal functions in weighted Bergman spaces can be approximated by polynomials, providing bounds on approximation errors and exploring the relationship between function smoothness and convergence rates.

## Contribution

It introduces new bounds on polynomial approximation of extremal functions in weighted Bergman spaces and analyzes the impact of function smoothness on convergence speed.

## Key findings

- Derived bounds on approximation errors in $A^p_\alpha$ and uniform norms.
- Established relations between Bergman modulus of continuity and polynomial approximation convergence.
- Provided insights into the rate of convergence of polynomial approximants.

## 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.