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Classification theorems for operators preserving zeros in a strip

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Abstract

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.

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Authors and Affiliations

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

    Petter Brändén & Matthew Chasse

Authors
  1. Petter Brändén
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  2. Matthew Chasse
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Correspondence to Petter Brändén.

Additional information

The first author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. The research is also supported by the Göran Gustafsson Foundation.

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Brändén, P., Chasse, M. Classification theorems for operators preserving zeros in a strip. JAMA 132, 177–215 (2017). https://doi.org/10.1007/s11854-017-0018-3

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  • DOI: https://doi.org/10.1007/s11854-017-0018-3

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