Quaternionic inverse Fourier transforms on locally compact abelian groups

Abstract

ABSTRACT In this paper, after introducing the concepts of quaternionic dual group and the quaternionic valued character on locally compact abelian group , the inverse of the quaternionic Fourier transform (QFT) on locally compact abelian groups is investigated. Due to the non-commutativity of multiplication of quaternions, there are different types of QFTs right, left and two-sided quaternionic Fourier transform. We focus on the right-sided quaternionic Fourier transform (RQFT) and two-sided quaternionic Fourier transform (SQFT). We establish the quaternionic Plancherel and inversion theorems for the square integrable quaternionic-valued signals on , the space , where G is a locally compact abelian group. Also RQFT on the space is studied. Furthermore relations between RQFT and SQFT are discussed. These results provide new proofs for the classical inverse Fourier transform, Plancherel theorem, etc. in .