# A planar large sieve and sparsity of time-frequency representations

## TL;DR

This paper extends the large sieve principle to time-frequency representations using a planar maximum Nyquist density, enabling perfect recovery and approximation of sparse signals in the modulation space via $L_1$-minimization.

## Contribution

It introduces the planar maximum Nyquist density and establishes a large sieve principle for time-frequency distributions with Gaussian windows, facilitating sparse signal recovery.

## Key findings

- Allows perfect recovery of the short-Fourier transform in $M_1$.
- Enables approximation of missing STFT data in $M_1$.
- Applies to signals corrupted by sparse noise.

## Abstract

With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, and used it to extend Bombieri's principle of the large sieve to bandlimited functions. This led to several recovery algorithms based on the minimization of the $L_{1}$-norm. In this paper we introduce the concept of {\ planar maximum} Nyquist density, which measures the sparsity of the time-frequency distribution of a function. We obtain a planar large sieve principle which applies to time-frequency representations with a gaussian window, or equivalently, to Fock spaces, $\mathcal{F}_{1}\left( \mathbb{C}\right) $, allowing for perfect recovery of the short-Fourier transform (STFT) of functions in the modulation space $M_{1}$ (also known as Feichtinger's algebra $S_{0}$) corrupted by sparse noise and for approximation of missing STFT data in $M_{1}$, by $L_{1}$-minimization.