Approximating z-bar in the Bergman space

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.

arXiv comment11 pages, 9 figures, to be submitted to the Proceedings of the 2015 conference "Completeness problems, Carleson measures and spaces of analytic functions" at the Mittag-Leffler institute