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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      PAGES = {ix+183},
      MRCLASS = {41.1X},
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      }
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      AUTHOR = {Obreschkoff, Nikola},
      TITLE = {Sur les fonctions entières limites de polynômes dont les zéros sont réel et entrelacés},
      JOURNAL = {Mat. Sbornik},
      FJOURNAL = {Rec. Math. [Mat. Sbornik] N.S.},
      VOLUME = {9 (51)},
      YEAR = {1941},
      PAGES = {421--428},
      MRCLASS = {30.0X},
      MRNUMBER = {2,356b},
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      ZBLNUMBER = {0024.32903},
      }
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      AUTHOR = {Obreschkoff, Nikola},
      TITLE = {Verteilung und {B}erechnung der {N}ullstellen Reeller {P}olynome},
      PUBLISHER = {VEB Deutscher Verlag der Wissenschaften},
      ADDRESS = {Berlin},
      YEAR = {1963},
      PAGES = {viii+298},
      MRCLASS = {30.11},
      MRNUMBER = {29 \#1302},
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      MRREVIEWER = {M. Marden},
      }
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    @incollection{Pin, MRKEY = {1924868},
      AUTHOR = {Pinkus, Allan},
      EDITOR = {Chui, C. K. and Schumaker, L. L. and Stoeckler, J.},
      TITLE = {Some remarks on zero-increasing transformations},
      BOOKTITLE = {Approximation Theory, {X}: Abstract and Classical Analysis},
      VENUE = {St. Louis, MO, 2001},
      SERIES = {Innov. Appl. Math.},
      PAGES = {333--352},
      PUBLISHER = {Vanderbilt Univ. Press},
      ADDRESS = {Nashville},
      YEAR = {2002},
      MRCLASS = {30C10 (30C15 41A10)},
      MRNUMBER = {2003f:30005},
      ZBLNUMBER = {1061.30006},
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      }
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      AUTHOR = {P{ó}lya, G.},
      TITLE = {Bemerkung Über die {I}ntegraldarstellung der {R}iemannschen {$\zeta$}-{F}unktion},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {48},
      YEAR = {1926},
      NUMBER = {3-4},
      PAGES = {305--317},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1555227},
      DOI = {10.1007/BF02565336},
      JFMNUMBER = {52.0335.02},
      }
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      author = {G. P{ó}lya},
      TITLE = {{Ü}ber {A}nnäherung durch {P}olynome mit lauter reellen {W}urzeln},
      JOURNAL = {Rend. Circ. Mat. Palermo},
      VOLUME = {36},
      YEAR = {1913},
      PAGES = {279--295},
      JFMNUMBER = {44.0474.01},
      }
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      author = {P{ó}lya, G. and Schur, I.},
      TITLE = {{Ü}ber zwei {A}rten von {F}aktorenfolgen in der {T}heorie der algebraischen {G}leichungen},
      JOURNAL = {J. Reine Angew. Math.},
      VOLUME = {144},
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      JFMNUMBER = {45.0176.01},
      }
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      AUTHOR = {P{ó}lya, G. and Szeg{ö},
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      TITLE = {Problems and Theorems in Analysis, {{\rm II}}: {T}heory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry},
      INVEDITION = {Translation of 4th German},
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      PUBLISHER = {Springer},
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      YEAR = {1976},
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      PAGES = {xiv+742},
      ISBN = {0-19-853493-0},
      MRCLASS = {30C10 (00A05 11C08 12D10 30C15 31-02 41A05)},
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      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {},
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      ISSN = {},
      }
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      DOI = {10.1090/S0002-9904-1943-07919-0},
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      FJOURNAL = {Mathematische Zeitschrift},
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      YEAR = {1922},
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      PAGES = {28--55},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {Contributed Item},
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      AUTHOR = {Walsh, J. L.},
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      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
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      YEAR = {1922},
      NUMBER = {3},
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      ISSN = {0002-9947},
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      MRCLASS = {12D10 (30C15)},
      MRNUMBER = {1501220},
      DOI = {10.2307/1989023},
      }

Keywords
, , , , ,
Mathematical Subject Classification
Primary: , , , , Secondary:
DOI
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