Umbrella theorems for time-frequency representations

Paolo Boggiatto; Chiara Boiti; Alessandro Oliaro

functional analysis

Umbrella theorems for time-frequency representations

arXiv:2607.14963

summary

The paper studies Umbrella Theorems—uncertainty principles that forbid a single square‑integrable function from dominating an infinite orthonormal family and its Fourier transforms—extending these results to higher‑dimensional L² spaces and to L² on the positive half‑line via the Mellin transform.

Abstract

An uncertainty principle due to H.S. Shapiro, the so-called Umbrella Theorem, asserts that there is no square integrable function uniformly dominating all the elements of an orthonormal family in $L^{2} (R)$ and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in $L^{2} (R^{d})$ related to time-frequency representations. We further extend the analysis to the case of $L^{2} (R^{+})$ , by means of the Mellin transform.

Topics & keywords

#uncertainty principle #time-frequency analysis #orthonormal families #fourier transform #mellin transform #harmonic analysisUmbrella TheoremL²(R^d)time-frequency representationMellin transformorthonormal basisuncertainty principle