Wavelet transform and Radon transform on the Quaternion Heisenberg group

TL;DR

This paper develops wavelet and Radon transform theories on the quaternion Heisenberg group, introducing radial wavelets, analyzing their properties, and providing inversion formulas using wavelet techniques.

Contribution

It introduces a unified framework for wavelet and Radon transforms on the quaternion Heisenberg group, including the construction of radial wavelets and their application to inversion formulas.

Findings

Constructed a class of radial wavelets on the quaternion Heisenberg group

Established the Radon transform as a bijection on a Semyanistri-Lizorkin space

Provided wavelet-based inversion formulas for the Radon transform

Abstract

Let $\mathscr Q$ be the quaternion Heisenberg group, and let $\mathbf P$ be the affine automorphism group of $\mathscr Q$. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of $\mathbf P$ on $L^2(\mathscr Q)$. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on $\mathscr Q$. A Semyanistri-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on $\mathscr Q$ both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth.