A Family of Unitary Operators Satisfying a Poisson-type Summation Formula

TL;DR

This paper introduces a family of unitary operators satisfying a weighted Poisson summation formula, characterizes their diagonal forms, and explores their relation to Fourier transforms and derivatives.

Contribution

It establishes the existence and uniqueness of such operators under various decay conditions and generalizes their connection to Fourier analysis.

Findings

Existence and uniqueness of unitary Fourier-Poisson operators.

Diagonalization of these operators.

Generalization of Fourier-derivative interplay.

Abstract

We consider a weighted form of the Poisson summation formula. We prove that under certain decay rate conditions on the weights, there exists a unique unitary Fourier-Poisson operator which satisfies this formula. We next find the diagonal form of this operator, and prove that under weaker conditions on the weights, a unique unitary operator still exists which satisfies a Poisson summation formula in operator form. We also generalize the interplay between the Fourier transform and derivative to those Fourier-Poisson operators.