Abstract:
For functions that take values in the Clifford algebra, we study the Clifford–Fourier transform on \mathbb R^m defined with a kernel function $K\ (x,\ y):=e^{\frac{{\ma...Show MoreMetadata
Abstract:
For functions that take values in the Clifford algebra, we study the Clifford–Fourier transform on \mathbb R^m defined with a kernel function K\ (x,\ y):=e^{\frac{{\mathrm i}\pi}{2}\Gamma_{\underline y}}\ {\mathrm e}^{-{\mathrm i}\langle\underline x,\underline y\rangle}, replacing the kernel e^{{\mathrm i}\langle\underline x,\underline y\rangle} of the ordinary Fourier transform, where \Gamma_{\underline y}:=-\Sigma_{j< k}e_je_k(y_j\partial_{y_k}-Y_k\partial_{y_j}). An explicit formula of K(x,y) is derived, which can be further simplified to a finite sum of Bessel functions when m is even. The closed formula of the kernel allows us to study the Clifford–Fourier transform and prove the inversion formula, for which a generalized translation operator and a convolution are defined and used.
Published in: International Mathematics Research Notices ( Volume: 2011, Issue: 22, 2011)
DOI: 10.1093/imrn/rnq288