Umbrella theorems for time-frequency representations

2026/07/16 by Paolo Boggiatto, Chiara Boiti, Alessandro Oliaro
Mathematics · #math.FA

paper · pdf

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 L²(ℝ) and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in L²(ℝ^d) related to time-frequency representations. We further extend the analysis to the case of L²(ℝ⁺), by means of the Mellin transform.

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