Poisson summation formula in weighted Lebesgue spaces

TL;DR

This paper characterizes parameter conditions in weighted Lebesgue spaces that ensure the Poisson summation formula holds, extending previous work by Kahane and Lemarié-Rieusset.

Contribution

It provides a complete characterization of parameters in weighted Lebesgue spaces guaranteeing the Poisson summation formula's validity.

Findings

Identifies parameter ranges for weighted Lebesgue spaces where the Poisson summation formula applies.

Completes the theoretical framework initiated by Kahane and Lemarié-Rieusset.

Advances understanding of Fourier analysis in weighted function spaces.

Abstract

We characterize the parameters $(\alpha,\beta,p,q)$ for which the condition $f|x|^\alpha\in L^p$ and $\widehat{f}|\xi|^\beta\in L^q$ implies the validity of the Poisson summation formula, thus completing the study of Kahane and Lemari\'e-Rieusset.