Model spaces results for the Gabor and Wavelet transforms

TL;DR

This paper characterizes the unique Gabor atom as the Gaussian and provides analogous results for wavelet transforms, offering new insights into irregular frames and improvements for specific classes of Gabor atoms and wavelets.

Contribution

It establishes the uniqueness of the Gaussian as the Gabor atom with an analytical model space and introduces a new approach to study irregular Gabor and wavelet frames.

Findings

Gaussian is the unique Gabor atom with an analytical model space

Provides an analogous uniqueness result for wavelet transforms

Improves results for Gabor atoms in the Feichtinger algebra and certain wavelets

Abstract

We prove that the unique Gabor atom with analytical model space is the Gaussian function. We give an analogous result for the wavelet transform. For the general case we give a new approach to study the irregular Gabor and wavelet frames. We improve some results for Gabor atoms in the Feichtinger algebra, and for a special class of wavelets.