Quantitative uncertainty principles for time-frequency Gaussian decay

Abstract

For real symmetric positive definite matrices $A$ and $B$, we characterize when a function $f\in L^2({\mathbb{R}}^d)$ satisfies \[ |f(x)| \lesssim e^{-(\frac12 - λ) \langle Ax, x\rangle} \quad \text{and} \quad |\widehat{f}(ξ)| \lesssim e^{-(\frac12 - λ) \langle Bξ, ξ\rangle} , \qquad \forall λ> 0 , \] 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.

36 pages