Clifford algebras,Fourier transforms,and quantum mechanics |
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Authors: | H. De Bie |
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Affiliation: | Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, , Galglaan 2, 9000 Ghent, Belgium |
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Abstract: | In this review, we give an overview of several recent generalizations of the Fourier transform, related to either the Lie algebra or the Lie superalgebra . In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform, and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford–Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel, and connection with various special functions. Copyright © 2012 John Wiley & Sons, Ltd. |
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Keywords: | generalized Fourier transform fractional Fourier transform Dunkl transform radially deformed Fourier transform super Fourier transform Clifford analysis Clifford– Fourier transform Hermite semigroup |
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