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The Bergman Analytic Content of Planar Domains

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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 the 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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Acknowledgements

We wish to thank Dmitry Khavinson for helpful discussions and valuable feedback. We would also like to acknowledge the contribution of Jan-Fredrik Olsen who suggested Theorem 1.2 as a conjecture during conversations at the conference “Completeness problems, Carleson measures, and spaces of analytic functions” at Mittag-Leffler in 2015.

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Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, USA

    Matthew Fleeman

  2. Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA

    Erik Lundberg

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  1. Matthew Fleeman
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  2. Erik Lundberg
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Correspondence to Erik Lundberg.

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Communicated by Eli Levin.

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Fleeman, M., Lundberg, E. The Bergman Analytic Content of Planar Domains. Comput. Methods Funct. Theory 17, 369–379 (2017). https://doi.org/10.1007/s40315-016-0189-4

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  • DOI: https://doi.org/10.1007/s40315-016-0189-4

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