# Summability properties of Gabor expansions

## TL;DR

This paper investigates the summability and spectral synthesis properties of Gabor systems generated by Gaussians, revealing limitations in their basis properties and the existence of summation methods.

## Contribution

It demonstrates that certain Gaussian Gabor systems are not strong Markushevich bases and establishes spectral synthesis with a one-dimensional defect.

## Key findings

- Existence of Gaussian Gabor systems that are not strong Markushevich bases.
- Spectral synthesis holds up to a one-dimensional defect for these systems.
- No linear summation method exists for general Gaussian Gabor expansions.

## Abstract

We show that there exist complete and minimal systems of time-frequency shifts of Gaussians in $L^2(\mathbb{R})$ which are not strong Markushevich basis (do not admit the spectral synthesis). In particular, it implies that there is no linear summation method for general Gaussian Gabor expansions. On the other hand we prove that the spectral synthesis for such Gabor systems holds up to one dimensional defect.