Quantitative uncertainty principles for time-frequency Gaussian decay

TL;DR

This work develops a comprehensive quantitative framework connecting time-frequency Gaussian decay with skewed Hermite expansions for $f \in L^2(\mathbb{R}^d)$. By using the skewed Bargmann transform, the authors translate decay conditions into growth estimates of entire functions and consequently obtain sharp, dimension-aware bounds on the skewed Hermite coefficients. They identify precise thresholds and dichotomies: when $|f(x)|$ and $|\widehat{f}(\xi)|$ are Gaussian with respect to $A$ and $A^{-1}$, the skewed Hermite coefficients decay at exponential rates; if $B\neq A^{-1}$, the situation becomes rigid with zero or dense subspaces depending on positivity of $B^{-1}-A$. The paper also develops a coordinate-wise variant and introduces quadratic interpolation for weight functions to determine when coordinate bounds suffice for full Hermite control, including explicit results for weights like $t^a$ and $(\log_+ t)^{1+a}$. These results yield fine-grained uncertainty principles that quantify how near-optimal time-frequency localization constrains the structure of $f$.

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.

Theorems & Definitions (94)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Lemma 2.1
• proof
• Example 2.2
• Definition 2.3
• Lemma 2.4
• Corollary 2.5