# Sur le principe d’incertitude pour les familles orthonormales de $L_{2}(R)$

### Anne de Roton

Université de Lorraine, Vandoeuvre-lès-Nancy, France### Bahman Saffari

Université Paris-Sud, Orsay, France### Harold S. Shapiro

Royal Institute of Technology, Stockholm, Sweden### Gérald Tenenbaum

Université de Lorraine, Vandoeuvre-lès-Nancy, France

## Abstract

A result of uncertainty principle type due to H.S. Shapiro states that, given an infinite orthonormal family of $L_{2}(R)$, 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 $L_{2}(R)$ such that all elements and all their Fourier transforms are uniformly dominated by a constant multiple of $r(x):=1+∣x∣ 1 $.

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 $L_{2}(R)$ 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.

## Cite this article

Anne de Roton, Bahman Saffari, Harold S. Shapiro, Gérald Tenenbaum, Sur le principe d’incertitude pour les familles orthonormales de $L_{2}(R)$. Enseign. Math. 62 (2016), no. 1/2, pp. 285–300

DOI 10.4171/LEM/62-1/2-15