A fractal uncertainty principle for the short-time Fourier transform and Gabor multipliers

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 L2(Rd), we obtain norm estimates for Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, which we also use to derive explicit upper bound asymptotes for the multidimensional Cantor iterates, in particular. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).