Abstract Harmonic Analysis on the General Affine Group GA(n,R)

TL;DR

This paper extends harmonic analysis by developing a Fourier transform and proving the Plancherel theorem for the general affine group, broadening the understanding of analysis on non-compact Lie groups.

Contribution

It generalizes the Fourier transform and establishes the Plancherel theorem specifically for the general affine group, a significant extension of harmonic analysis techniques.

Findings

Established the Fourier transform on the general affine group.

Proved the Plancherel theorem for this group.

Extended harmonic analysis tools to a broader class of groups.

Abstract

The general linear group has two components and its the identity component, which consists of the real matrices with positive determinant and the set of all matrices with negative determinant. Since the general linear group is a two copies of the group of the identity component, so the general affine group. Consider the affine group, which is the semidirect product of the identity component with the real group of dimension. In this paper we generalize the Fourier transform to obtain the Plancherel theorem on this group and then, we establish the Plancherel theorem for the general affine group