Pólya-Schur master theorems for circular domains and their boundaries

Pages 465-492 from Volume 170 (2009), Issue 1 by Julius Borcea, Petter Brändén

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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      VENUE = {St. Louis, MO, 2001},
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      FJOURNAL = {Acta Mathematica},
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      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {Contributed Item},
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Keywords
, , , , ,
Mathematical Subject Classification
Primary: , , , , Secondary:
DOI
https://doi.org/10.4007/annals.2009.170.465
MR
2521123
zbMATH
1184.30004
Milestones
Received: 3 September 2006
Published online: 12 July 2009

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