Heisenberg's and Hardy's Uncertainty Principles in Real Clifford Algebras

TL;DR

This paper extends classical uncertainty principles, specifically Heisenberg's inequality and Hardy's theorem, to the setting of real Clifford algebras, broadening their mathematical scope.

Contribution

It introduces analogues of key uncertainty principles within the framework of the real Clifford Fourier transform, a recent development in Clifford analysis.

Findings

Established Clifford algebra versions of Heisenberg's inequality

Proved Hardy's theorem in the context of real Clifford algebras

Enhanced understanding of Fourier analysis in Clifford algebra settings

Abstract

Recently, many surveys are devoted to study the Clifford Fourier transform. Dealing with the real Clifford Fourier transform introduced by Hitzer [10], we establish analogues of the classical Heisenberg's inequality and Hardy's theorem in the real Clifford algebra Cl(p, q).