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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Wronskians of Fourier and Laplace transforms
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by Dimitar K. Dimitrov and Yuan Xu PDF
Trans. Amer. Math. Soc. 372 (2019), 4107-4125 Request permission

Abstract:

Associated with a given suitable function, or a measure, on $\mathbb {R}$, we introduce a correlation function so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on $\mathbb {R}$ are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on $\mathbb {R}_+$ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel $K$, belongs to the Laguerre–Pólya class, which answers an old question of Pólya. The characterization is given in terms of a density property of the correlation kernel related to $K$, via classical results of Laguerre and Jensen and employing Wiener’s $L^1$ Tauberian theorem. As a consequence, we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann $\xi$-function.
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Additional Information
  • Dimitar K. Dimitrov
  • Affiliation: Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista, 15054-000 Saõ José do Rio Preto, São Paulo, Brazil
  • MR Author ID: 308699
  • Email: d\string_k\string_dimitrov@yahoo.com
  • Yuan Xu
  • Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
  • MR Author ID: 227532
  • Email: yuan@uoregon.edu
  • Received by editor(s): July 28, 2017
  • Received by editor(s) in revised form: July 31, 2018
  • Published electronically: June 3, 2019
  • Additional Notes: The first author was supported by the Brazilian foundations CNPq, under Grant 306136/2017-1, and FAPESP, under Grants 2016/09906-0 and 2014/08328-8.
    The second author was supported in part by NSF Grant DMS-1510296
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4107-4125
  • MSC (2010): Primary 11M26, 42B10; Secondary 30D10, 30D15, 40E05, 42A82
  • DOI: https://doi.org/10.1090/tran/7809
  • MathSciNet review: 4009426