# Helgason Gabor Fourier transform and uncertainty principles

## TL;DR

This paper generalizes the classical Gabor-Fourier transform to Riemannian symmetric spaces, introduces the Helgason Gabor Fourier transform, and proves key properties and an uncertainty principle for it.

## Contribution

It introduces the Helgason Gabor Fourier transform on symmetric spaces and establishes its fundamental properties and uncertainty principles, extending classical Fourier analysis.

## Key findings

- Established the reconstruction formula for HGFT
- Proved the Plancherel and Parseval formulas for HGFT
- Derived a local uncertainty principle similar to Benedicks' principle

## Abstract

Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where ceratin frequencies occur in the input signal, this method is introduced by Dennis Gabor. In this paper, we generalize the classical Gabor-Fourier transform(GFT) to the Riemannian symmetric space called the Helgason Gabor Fourier transform (HGFT). We continue with proving several important properties of HGFT, like the reconstruction formula, the Plancherel formula, and Parseval formula. Finally we establish some local uncertainty principle such as Benedicks-type uncertainty principle