Continuous Gabor transform for a class of non-Abelian groups

TL;DR

This paper extends the continuous Gabor transform to certain non-Abelian groups, providing formulas for inversion and Plancherel, with applications to the Heisenberg and SL(2,R) groups.

Contribution

It introduces a new continuous Gabor transform framework for non-Abelian groups, including inversion and Plancherel formulas, with practical examples.

Findings

Derived a Plancherel formula for the new transform

Established an inversion formula for the transform

Applied the framework to the Heisenberg and SL(2,R) groups

Abstract

In this article we define the continuous Gabor transform for second countable, non-abelian, unimodular and type I groups and also we investigate a Plancherel formula and an inversion formula for our definition. As an example we show that how these formulas work for the Heisenberg group and also the matrix group ${SL(2,\mathbb{R})}$.