WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS
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Data
2019-09-15
Autores
Dimitrov, Dimitar K. [UNESP]
Xu, Yuan
Título da Revista
ISSN da Revista
Título de Volume
Editor
Amer Mathematical Soc
Resumo
Associated with a given suitable function, or a measure, on 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 non-negative function on R are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on 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-Polya class, which answers an old question of Polya. 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.
Descrição
Palavras-chave
Fourier transform, Laplace transform, Wronskian, entire function, Laguerre-Polya class, Riemann hypothesis
Como citar
Transactions Of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 372, n. 6, p. 4107-4125, 2019.