The Quaternionic Affine Group and Related Continuous Wavelet Transforms on Complex and Quaternionic Hilbert Spaces

TL;DR

This paper develops the theory of quaternionic affine groups and constructs continuous wavelet transforms on complex and quaternionic Hilbert spaces, extending wavelet analysis to quaternionic settings.

Contribution

It introduces unitary irreducible representations of the quaternionic affine group on complex and quaternionic Hilbert spaces, enabling quaternionic wavelet transforms.

Findings

Representations are square-integrable.

Constructed quaternionic wavelets and transforms.

Extended wavelet analysis to quaternionic spaces.

Abstract

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilbert space. As in the real and complex cases, the representations for the quaternionic group also turn out to be square-integrable. Using these representations we constrct quaternionic wavelets and continuous wavelet transforms on both the complex and quaternionic Hilbert spaces.