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Hyponormal Quantization of Planar Domains pp 77–92Cite as

Mother Bodies

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2199))

Abstract

We outline the general theory for a certain kind of potential theoretic skeletons, or ‘mother bodies’, associated to a given domain. The hope is, generally speaking, that such skeletons can be identified as attractors for zeros of orthogonal polynomials, and in a few cases such expectations have indeed been met, theoretically and/or experimentally. For the exponential polynomials the success is rather limited so far, but by building in enough flexibility in the models one expects in some not so distant future to reach a reasonable matching. In the present chapter we set up desirable properties (formulated as ‘axioms’) to be satisfied by mother bodies. Since the search for potential theoretic skeletons is a highly ill-posed problem (related to the Cauchy problem for an elliptic operator) very few domains admit mother bodies, but for domains with piecewise algebraic boundaries there is a rather constructive and efficient theory, bearing in mind that the same class of domains is also amenable for studying zeros of orthogonal polynomials.

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Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

    Björn Gustafsson

  2. Mathematics Department, University of California, Santa Barbara, CA, USA

    Mihai Putinar

  3. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle Upon Tyne, UK

    Mihai Putinar

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  1. Björn Gustafsson
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Gustafsson, B., Putinar, M. (2017). Mother Bodies. In: Hyponormal Quantization of Planar Domains. Lecture Notes in Mathematics, vol 2199. Springer, Cham. https://doi.org/10.1007/978-3-319-65810-0_6

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