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arXiv · 2005.02996

Fourier interpolation with zeros of zeta and $L$-functions

Andriy Bondarenko·Danylo Radchenko·Kristian Seip
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Abstract

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

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Andriy Bondarenko, Danylo Radchenko, Kristian Seip. 2022-11-03. Fourier interpolation with zeros of zeta and $L$-functions. https://arxiv.org/abs/2005.02996

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