Two-sided Clifford-valued Linear Canonical Transform: Properties and Mustard Convolution

The present study is the first of its kind which aims to anal-yse the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued linear canonical transform in L-2(R-n, Cl-0,Cl-n), which not only embodies the classical Clifford-Fourier transform, but also yields another new variant of Clifford transforms based on the frac-tional Clifford-Fourier transform. To begin with, we study all funda-mental properties of the proposed transform, including the inversion formula, translation and scaling covariances, Plancherel and differentia-tion theorems. Subsequently, we introduce a novel Clifford-valued Mus-tard convolution associated with the proposed transform and express the proposed convolution in terms of linear combination of eight standard convolutions.