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arXiv · 2607.14963

Umbrella theorems for time-frequency representations

Paolo Boggiatto·Chiara Boiti·Alessandro Oliaro
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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 $L^2(\mathbb{R})$ and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in $L^2(\mathbb{R}^d)$ related to time-frequency representations. We further extend the analysis to the case of $L^2(\mathbb{R}^+)$, by means of the Mellin transform.

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Paolo Boggiatto, Chiara Boiti, Alessandro Oliaro. 2026-07-16. Umbrella theorems for time-frequency representations. https://arxiv.org/abs/2607.14963

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