A generalization of the Beurling-Hedenmalm uncertainty principle

Abstract

We study the uncertainty principles of Hardy and of Beurling, and functions that "only just" satisfy the inequalities of uncertainty principles. More specifically, we show that if a function and its Fourier transform have nearly gaussian decay, the the coefficients of its Hermite expansion decay fast, and vice versa. We give a new and simple proof of generalisation of Beurling's uncertainty principle first in R using complex analysis. Then we generalise to R^n using various techniques. Also we illustrate connections with the classical moment problem.