Abstract
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.
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Communicated by Uwe Kaehler.
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Teali, A.A., Shah, F.A. Two-sided Clifford-valued Linear Canonical Transform: Properties and Mustard Convolution. Adv. Appl. Clifford Algebras 33, 18 (2023). https://doi.org/10.1007/s00006-023-01266-y
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DOI: https://doi.org/10.1007/s00006-023-01266-y
Keywords
- Clifford-valued linear canonical transform
- Clifford-valued Fourier transform
- Mustard convolution
- Differentiation theorem