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Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

Williams, Cameron L. ; Bodmann, Bernhard G. ; Kouri, Donald J.

The Journal of fourier analysis and applications, 2017-06, Vol.23 (3), p.660-678 [Periódico revisado por pares]

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Título:
Fourier and Beyond: Invariance Properties of a Family of Integral Transforms
Autor: Williams, Cameron L. ; Bodmann, Bernhard G. ; Kouri, Donald J.
Assuntos: Abstract Harmonic Analysis ; Approximations and Expansions ; Eigenvalues ; Eigenvectors ; Fourier Analysis ; Fourier transforms ; Integral transforms ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Orthogonality ; Partial Differential Equations ; Real numbers ; Signal,Image and Speech Processing
É parte de: The Journal of fourier analysis and applications, 2017-06, Vol.23 (3), p.660-678
Descrição: The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation-related property and by having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.
Editor: New York: Springer US
Idioma: Inglês

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