Pólya-Schur master theorems for circular domains and their boundaries

Abstract

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region $\Omega\subseteq \mathbb{C}$ for arbitrary closed circular domains $\Omega$ (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for $\Omega=\mathbb{R}$ our results settle open questions that go back to Laguerre and Pólya-Schur.

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      JOURNAL = {Mat. Sbornik},
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      VOLUME = {9 (51)},
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      PAGES = {421--428},
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      AUTHOR = {Obreschkoff, Nikola},
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      PUBLISHER = {VEB Deutscher Verlag der Wissenschaften},
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      VENUE = {St. Louis, MO, 2001},
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      PAGES = {333--352},
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      NUMBER = {3-4},
      PAGES = {305--317},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {Contributed Item},
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      MRNUMBER = {1501220},
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Authors

Julius Borcea

Department of Mathematics
Stockholm University
SE-106 91 Stockholm
Sweden

Petter Brändén

Department of Mathematics
Royal Institute of Technology
SE-100 44 Stockholm
Sweden