Abstract
In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.
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Acknowledgments
We thank the referees for their valued comments which significantly improved the clarity of the paper. H. De Bie is supported by the UGent BOF starting Grant 01N01513. P. Lian is supported by the scholarship from Chinese Scholarship Council (CSC) under the CSC No. 201406120169.
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Communicated by Hans G. Feichtinger.
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Constales, D., De Bie, H. & Lian, P. A new construction of the Clifford-Fourier kernel. J Fourier Anal Appl 23, 462–483 (2017). https://doi.org/10.1007/s00041-016-9476-8
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DOI: https://doi.org/10.1007/s00041-016-9476-8