The present study is the first of its kind which aims to analyse the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued linear canonical transform in \(L^2({\mathbb {R}}^n, C\ell _{0,n})\), which not only embodies the classical Clifford–Fourier transform, but also yields another new variant of Clifford transforms based on the fractional Clifford–Fourier transform. To begin with, we study all fundamental properties of the proposed transform, including the inversion formula, translation and scaling covariances, Plancherel and differentiation theorems. Subsequently, we introduce a novel Clifford-valued Mustard convolution associated with the proposed transform and express the proposed convolution in terms of linear combination of eight standard convolutions.

本研究是同类研究中的第一项，旨在通过在\(L^2({\mathbb {R}}^n, C\ell _{0,n})\)，它不仅体现了经典的克利福德-傅里叶变换，而且在分数阶克利福德-傅里叶变换的基础上产生了另一种新的克利福德变换变体。首先，我们研究了所提出变换的所有基本属性，包括反演公式、平移和缩放协方差、Plancherel 和微分定理。随后，我们介绍了一种与所提出的变换相关的新型克利福德值 Mustard 卷积，并根据八个标准卷积的线性组合来表达所提出的卷积。