A result of uncertainty principle type due to H.S. Shapiro states that, given an infinite orthonormal family of , 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 such that all elements and all their Fourier transforms are uniformly dominated by a constant multiple of .
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 all of whose elements and all their Fourier transforms are dominated at infinity by a function with decay strictly faster than , 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 . Enseign. Math. 62 (2016), no. 1/2, pp. 285–300DOI 10.4171/LEM/62-1/2-15