Translation-Invariant Clifford Operators

Abstract

This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space $L^2(\mathbb{R}^2, \mathbb{H})$ of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of $L^2(\mathbb{R}^2, \mathbb{H})$ and its bounded linear translation-invariant operators. The Clifford-Fourier characterisation of Hardy-type spaces on $\mathbb{R}^d$ is also explored.