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Fourier Transforms of Positive Definite Kernels and the Riemann \(\xi \)-Function

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Abstract

The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between the Bochner–Khinchin–Mathias theory of positive definite kernels and the generalized real Laguerre inequalities. The concavity and convexity properties of the Jacobi theta function play a prominent role throughout this work. The paper concludes with several questions and open problems.

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  1. Department of Mathematics, University of Hawaii, Honolulu, HI, 96822, USA

    George Csordas

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  1. George Csordas
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Correspondence to George Csordas.

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Communicated by Serguei Shimorin.

Dedicated to Dr. Robert D. Cushnie on the occasion of his $$80$$ 80 th birthday.

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Csordas, G. Fourier Transforms of Positive Definite Kernels and the Riemann \(\xi \)-Function. Comput. Methods Funct. Theory 15, 373–391 (2015). https://doi.org/10.1007/s40315-014-0105-8

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