Hilbert Transformation and $r\mathrm{Spin}(n)+\mathbb{R}^n$ Group

TL;DR

This paper investigates the symmetry properties of the Hilbert transformation in multiple variables within the Clifford algebra framework, introducing a new group extension and explicitly characterizing the transformation via spinor representations for dimensions 2 and 3.

Contribution

It introduces the group rSpin(n)+R^n as an extension of the ax+b group and characterizes the Hilbert transformation's symmetry properties using spinor representations in low dimensions.

Findings

Hilbert transformation exhibits specific symmetry under rSpin(n)+R^n.

Explicit spinor representations for n=2,3 are derived.

The Hilbert transformation is characterized by invariance under the group action.

Abstract

In this paper we study symmetry properties of the Hilbert transformation of several real variables in the Clifford algebra setting. In order to describe the symmetry properties we introduce the group $r\mathrm{Spin}(n)+\mathbb{R}^n, r>0,$ which is essentially an extension of the ax+b group. The study concludes that the Hilbert transformation has certain characteristic symmetry properties in terms of $r\mathrm{Spin}(n)+\mathbb{R}^n.$ In the present paper, for $n=2$ and $3$ we obtain, explicitly, the induced spinor representations of the $r\mathrm{Spin}(n)+\mathbb{R}^n$ group. Then we decompose the natural representation of $r\mathrm{Spin}(n)+\mathbb{R}^n$ into the direct sum of some two irreducible spinor representations, by which we characterize the Hilbert transformation in $\mathbb{R}^3$ and $\mathbb{R}^2.$ Precisely, we show that a nontrivial skew operator is the Hilbert transformation if and only if it is invariant under the action of the $r\mathrm{Spin}(n)+\mathbb{R}^n, n=2,3,$ group.