A Fractal Uncertainty Principle for the Short-Time Fourier Transform and Gabor multipliers

TL;DR

This paper establishes a fractal uncertainty principle for the Short-Time Fourier Transform with Gaussian windows, providing new bounds for time-frequency localization and implications for Gabor multipliers.

Contribution

It introduces a fractal uncertainty principle in the time-frequency domain, with explicit bounds and applications to Gabor multipliers, extending prior uncertainty principles.

Findings

Norm estimates for Daubechies' localization operator on porous sets

Explicit bounds for multidimensional Cantor sets

Translation of the principle to discrete Gabor multipliers

Abstract

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, and for multidimensional Cantor iterates we derive explicit upper bound asymptotes. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.