Quantitative uncertainty principles for time-frequency Gaussian decay

TL;DR

This paper characterizes when functions and their Fourier transforms exhibit Gaussian decay in time-frequency space, using skewed Hermite series expansions, and explores conditions under which coordinate-wise decay aligns with these bounds.

Contribution

It provides a new characterization of Gaussian decay in functions and their Fourier transforms via skewed Hermite series, extending uncertainty principles.

Findings

Characterization of Gaussian decay using skewed Hermite series.

Conditions for equivalence between coordinate-wise decay and Hermite coefficient bounds.

Extension of uncertainty principles to time-frequency decay estimates.

Abstract

For real symmetric positive definite matrices $A$ and $B$, we characterize when a function $f \in L^2(\mathbb{R}^d)$ satisfies \[ |f(x)| \lesssim e^{-(\frac12 - \lambda) \langle Ax, x\rangle} \quad \text{and} \quad |\widehat{f}(\xi)| \lesssim e^{-(\frac12 - \lambda) \langle B\xi, \xi\rangle} , \qquad \forall \lambda > 0 , \] or even more specified time-frequency decay estimates, in terms of the skewed Hermite series expansion of $f$. We also consider coordinate-wise time-frequency decay and determine when it becomes equivalent to the same bounds on the skewed Hermite coefficients.