Sur le principe d’incertitude pour les familles orthonormales de $L^2(\mathbb{R})$

Abstract

A result of uncertainty principle type due to H.S. Shapiro states that, given an infinite orthonormal family of $L^2(\mathbb{R})$, there is no square integrable function uniformly dominating all functions and also all their Fourier transforms. However, Shapiro conjectured the existence of an orthonormal basis of $L^2(\mathbb{R})$ such that all elements and all their Fourier transforms are uniformly dominated by a constant multiple of $r(x):=\frac{1}{\sqrt{1+|x|}}$.

In this work, we provide a proof of Shapiro's uncertainty principle and we confirm his conjecture in a strong form, where one of the two upper bounds is replaced by a function with arbitrarily fast decay. We also show that, for a certain, natural type of basis, the initial bound is optimal. Finally, we construct an orthonormal family of $L^2(\mathbb{R})$ all of whose elements and all their Fourier transforms are dominated at infinity by a function $s(x)$ with decay strictly faster than $r(x)$, but which is not square-integrable in a neighbourhood of the origin.