Diffusive wavelets on the Spin group

TL;DR

This paper develops Clifford-valued diffusive wavelets on the Spin group by exploring its representation theory, modified diffusion equations, and eigenfunctions of the Laplace-Beltrami operator, advancing wavelet analysis on this mathematical structure.

Contribution

It introduces Clifford-valued diffusive wavelets on Spin(m) using a modified diffusion operator and characterizes all representations and eigenfunctions of relevant operators on Spin(m).

Findings

Constructed Clifford-valued diffusion wavelets on Spin(m).

Characterized all representations and eigenfunctions of the Casimir operator.

Derived the series expansion of the heat kernel on Spin(m).

Abstract

The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford analysis. In the second part we are interested in Clifford-valued functions and wavelets on the sphere. The connection of representations of Spin(m) and the concept of diffusive wavelets leads naturally to investigations of a modified diffusion equation on the sphere, that makes use of the Gamma operator. We will achieve to obtain Clifford-valued diffusion wavelets with respect to a modified diffusion operator. Since we are able to characterize all representations of Spin(m) and even to obtain all eigenvectors of the (by representation) regarded Casimir operator in representation spaces, it seems appropriate to look at functions on Spin(m) directly. Concerning this, our aim shall be to formulate eigenfunctions for the Laplace-Beltrami operator on Spin(m) and give the series expansion of the heat kernel on Spin(m) in terms of eigenfunctions.