Approximating z-bar in the Bergman space

TL;DR

This paper investigates the best approximation of the conjugate variable in the Bergman space, linking it to the Dirichlet problem solution and deriving inequalities related to domain geometry.

Contribution

It establishes a novel connection between the best approximation of ar in Bergman space and the Dirichlet problem, providing explicit examples and geometric inequalities.

Findings

Best approximation is the derivative of the Dirichlet problem solution.

Examples include polynomial and rational function approximations.

Derived the isoperimetric inequality for domain-related approximation distances.

Abstract

We consider the problem of finding the best approximation to $\bar{z}$ in the Bergman Space $A^{2}(\Omega)$. We show that this best approximation is the derivative of the solution to the Dirichlet problem on $\partial\Omega$ with data $\left|z\right|^{2}$ and give examples of domains where the best approximation is a polynomial, or a rational function. Finally, we obtain the "isoperimetric sandwich" for $dist(\overline{z},\Omega)$ that yields the celebrated St. Venant inequality for torsional rigidity.