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

TL;DR

This paper develops a new family of Fourier interpolation bases using zeros of zeta and L-functions, revealing duality principles and functional equations related to Dirichlet series and modular forms.

Contribution

It introduces a novel construction of Fourier bases based on zeros of zeta and L-functions, extending the theory with duality and functional equations.

Findings

Constructed Fourier bases involving zeros of zeta and L-functions.

Established duality principles for these bases.

Analyzed Dirichlet series kernels with functional equations.

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.