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Zeroes of Gaussian analytic functions with translation-invariant distribution

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Abstract

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the horizontal limiting measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the limiting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic functions with symmetry around the real axis. These results extend a work by Norbert Wiener.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Naomi D. Feldheim

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  1. Naomi D. Feldheim
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Correspondence to Naomi D. Feldheim.

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Research supported by the Science Foundation of the Israel Academy of Sciences and Humanities, grant 171/07.

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Feldheim, N.D. Zeroes of Gaussian analytic functions with translation-invariant distribution. Isr. J. Math. 195, 317–345 (2013). https://doi.org/10.1007/s11856-012-0130-0

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  • DOI: https://doi.org/10.1007/s11856-012-0130-0

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