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arXiv · 1602.03615

The Bergman analytic content of planar domains

Matthew Fleeman·Erik Lundberg
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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.

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Matthew Fleeman, Erik Lundberg. 2016-02-11. The Bergman analytic content of planar domains. https://arxiv.org/abs/1602.03615

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