Donoho-Logan Large Sieve Principles for Modulation and Polyanalytic Fock Spaces

TL;DR

This paper develops large sieve inequalities for the short-time Fourier transform in modulation spaces, enabling deterministic signal reconstruction from localized time-frequency data, with applications to polyanalytic Fock spaces and Gabor systems.

Contribution

It introduces a novel method combining Schur's test and Seip's formula to establish large sieve inequalities for the STFT with arbitrary localization sets.

Findings

Derived explicit large sieve constant estimates.

Provided a reconstruction formula from STFT values on arbitrary discs.

Extended results to polyanalytic Fock spaces and discrete Gabor systems.

Abstract

We obtain estimates for the $L^{p}$-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of $\mathbb{R}^{2}$, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we will obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions $T\times W$ of the time-frequency plane using the Fourier transform to intertwine time and frequency, we will localize the representation of the signals in terms of the short-time Fourier transform in sets $\Delta $ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for $f\in L^{2}(\mathbb{R})$ from its STFT values on arbitrary discs. A discussion on optimality follows, along the lines of Donoho-Stark paper on uncertainty principles and signal recovery. We also consider the case of discrete Gabor systems, vector-valued STFT transforms and rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.