Principe d'Heisenberg et fonctions positives

TL;DR

This paper explores the properties of Fourier transform pairs with positivity constraints, establishing a positive lower bound for the product of interval bounds in a number theory context, including multi-dimensional cases.

Contribution

It introduces a new problem linking Fourier analysis and number theory, analyzing bounds for Fourier pairs with specific positivity and negativity conditions.

Findings

Existence of a positive lower bound for the product of interval bounds

Extension of the problem to multiple dimensions

Connection between Fourier properties and number theory

Abstract

Starting from a problem in number theory, the article investigates the properties of the couples of Fourier transforms on the real line, f and g, real and even, f >= 0 out of an interval (-a, a) and f(0) < 0, g >= 0 out of an interval (-b, b) and g(0) < 0 . How small the product ab can be ? There is a strictly positive lower bound, the exact value is not known. The same problem is considered in several dimensions (where it is related to number theory, as the article points out).