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Transactions of the American Mathematical Society

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

 
 

 

Wronskians of Fourier and Laplace transforms


Authors: Dimitar K. Dimitrov and Yuan Xu
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11M26, 42B10; Secondary 30D10, 30D15, 40E05, 42A82
DOI: https://doi.org/10.1090/tran/7809
Published electronically: June 3, 2019
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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
Email: d\string_k\string_dimitrov@yahoo.com

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: yuan@uoregon.edu

DOI: https://doi.org/10.1090/tran/7809
Keywords: Fourier transform, Laplace transform, Wronskian, entire function, Laguerre--P\'olya class, Riemann hypothesis
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
Article copyright: © Copyright 2019 American Mathematical Society