Crystalline temperate distributions with uniformly discrete support and spectrum

TL;DR

This paper characterizes certain temperate distributions with uniformly discrete support and spectrum, showing they can be constructed from Poisson's summation formula through a finite sequence of basic operations.

Contribution

It provides a structural description of temperate distributions with discrete support and spectrum, linking them to Poisson's summation formula and basic operations.

Findings

Distributions with discrete support and spectrum can be generated from Poisson's formula.

Finite operations like shifts, modulations, and differentiations suffice.

The result bridges distribution theory and classical harmonic analysis.

Abstract

We prove that a temperate distribution on $\mathbb{R}$ whose support and spectrum are uniformly discrete sets, can be obtained from Poisson's summation formula by a finite number of basic operations (shifts, modulations, differentiations, multiplication by polynomials, and taking linear combinations).