Three Dimensional Sums of Character Gabor Systems

TL;DR

This paper addresses the square-root bottleneck in deterministic compressive sensing by analyzing a Gabor system of Legendre symbols, providing partial solutions to improve sparse signal recovery.

Contribution

It introduces a new approach using Gabor systems of Legendre symbols to partially overcome the square-root bottleneck in deterministic compressive sensing.

Findings

Provides a nontrivial upper bound for sums over consecutive vectors

Offers a partial solution to the square-root bottleneck problem

Enhances understanding of Gabor systems in sparse recovery

Abstract

In deterministic compressive sensing, one constructs sampling matrices that recover sparse signals from highly incomplete measurements. However, the so-called square-root bottleneck limits the usefulness of such matrices, as they are only able to recover exceedingly sparse signals with respect to the matrix dimension. In view of the flat restricted isometry property (flat RIP) proposed by Bourgain et al., we provide a partial solution to the bottleneck problem with the Gabor system of Legendre symbols. When summing over consecutive vectors, the estimate gives a nontrivial upper bound required for the bottleneck problem.