Fourier analysis on the affine group, quantization and noncompact Connes geometries

TL;DR

This paper develops a Fourier analysis framework on the affine group, introducing a quantizer and noncommutative product that connect signal processing and noncompact Connes geometries.

Contribution

It introduces a Stratonovich-Weyl quantizer for the affine group and constructs a noncommutative product linking signal analysis with noncompact spectral triples.

Findings

Derived a noncommutative product on the half-plane

Reproduced time-frequency distributions in signal processing

Extended Fourier transformations for the affine group

Abstract

We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.