Decay of coefficients and approximation rates in Gabor Gaussian frames

TL;DR

This paper provides an elementary, self-contained proof that functions can be approximated by Gaussian coherent states with the number of terms depending on their smoothness and decay properties.

Contribution

It offers a simple, self-contained proof of approximation rates in Gabor Gaussian frames, avoiding advanced modulation space theory.

Findings

Approximation rate depends on function smoothness and decay

Elementary proof technique used

Applicable to functions with specific decay and smoothness

Abstract

The aim of this note is to present a self-contained proof of the fact that a function can be approximated using a linear combination of Gaussian coherent states, with a number of terms controlled in terms of the smoothness and of the decay at infinity of the function. This result can easily be obtained using advanced results on modulation spaces, but the proof presented here is completely elementary and self-contained.