Fourier interpolation and time-frequency localization

Kulikov, Aleksei

doi:10.48550/arxiv.2005.12836

Cited by 1 publication

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“…This observation yields a precise notion of duality between the two sequences involved in a Fourier interpolation basis, closely aligned with modular relations as for example studied in some generality by Bochner [3]. Second, we discuss a recent density theorem of Kulikov [20], which is a version of the uncertainty principle valid for Fourier interpolation bases. We observe that there is a precise correspondence between Kulikov's density condition and the Riemann-von Mangoldt formula for the density of the nontrivial zeros of zeta and L-functions.…”

Section: Introductionmentioning

confidence: 60%

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

Bondarenko¹,

Radchenko²,

Seip³

2020

Preprint

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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. Contents 1. Introduction Outline of the paper 2. Generalities on Fourier interpolation 2.1. The Dirichlet series kernel associated with Λ 2.2. The Dirichlet series kernel associated with Λ * 2.3. Examples 2.4. The joint density of Λ and Λ * 2.5. Fourier interpolation and crystalline measures 3. Modular integrals for the theta group 3.1. Preliminaries on the theta group 3.2. Modular forms for the theta group 3.3. Modular kernels 3.4. Definition and basic properties of F ± k (τ, s) 4. The Dirichlet series kernel associated with zeros of ζ(s) 4.1. The Mellin transform of F ± k (τ, s) 4.2. Construction of the Dirichlet series kernels 4.3. Proof of Theorem 1.1 4.4. Relation with the Riemann-Weil formula 5. Fourier interpolation with zeros of Dirichlet L-functions and other Dirichlet series 2010 Mathematics Subject Classification. 11M06, 11F37, 42A10.

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Section: Introductionmentioning

confidence: 60%