The Bergman analytic content of planar domains

Abstract

Given a planar domain $\Omega$, the Bergman analytic content measures the $L^2(\Omega )$-distance between $\bar{z}$ and the Bergman space $A^2(\Omega )$. We compute the Bergman analytic content of simply-connected quadrature domains with quadrature formula supported at one point, and we also determine the function $f\in A^2(\Omega )$ that best approximates $\bar{z}$. We show that, for simply-connected domains, the square of Bergman analytic content is equivalent to torsional rigidity from classical elasticity theory, while for multiply-connected domains these two domain constants are not equivalent in general.

10 pages, 3 figures