Umbrella theorems for time-frequency representations

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.