On accumulated spectrograms for Gabor frames

TL;DR

This paper extends classical results on accumulated spectrograms to Gabor multipliers, demonstrating bounds on their approximation to mask indicators and showing hyperuniformity in certain Gabor frame ensembles.

Contribution

It introduces new bounds for accumulated spectrograms of Gabor multipliers and proves hyperuniformity of Weyl-Heisenberg ensembles on lattices.

Findings

Bound on lattice $ ext{ell}^1$ distance is sharp.

Accumulated spectrogram approximates mask indicator within boundary proximity.

Weyl-Heisenberg ensemble is hyperuniform on a lattice.

Abstract

Analogs of classical results on accumulated spectrograms, the sum of spectrograms of eigenfunctions of localization operators, are established for Gabor multipliers. We show that the lattice $\ell^1$ distance between the accumulated spectrogram and the indicator function of the Gabor multiplier mask is bounded by the number of lattice points near the boundary of the mask and that this bound is sharp in general. The methods developed for the proofs are also used to show that the Weyl-Heisenberg ensemble restricted to a lattice is hyperuniform when the Gabor frame is tight.