(k,a)-generalized wavelet transform and applications

We introduce the notion of the ( k ,  a )-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl wavelet transforms. The restriction of the ( k ,  a )-generalized wavelet transform to radial functions is given by the generalized Hankel w...

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Bibliographic Details
Published in:Journal of pseudo-differential operators and applications Vol. 11; no. 1; pp. 55 - 92
Main Author: Mejjaoli, Hatem
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.03.2020
Springer Nature B.V
Subjects:
Algebra
Analysis
Applications of Mathematics
Estimating techniques
Fourier transforms
Functional Analysis
Harmonic analysis
Localization
Mathematics
Mathematics and Statistics
Operator Theory
Operators (mathematics)
Partial Differential Equations
Regularization
Sobolev space
Wavelet transforms
Localization operators
generalized wavelet
generalized
Secondary 42B10
47G30
generalized Fourier
Theory of reproducing kernels
Tikhonov regularization
Primary 47G10
Laguerre semigroup
ISSN:1662-9981, 1662-999X
Online Access:Get full text
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Summary:We introduce the notion of the ( k ,  a )-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl wavelet transforms. The restriction of the ( k ,  a )-generalized wavelet transform to radial functions is given by the generalized Hankel wavelet transform. We prove for this new transform Plancherel’s formula, inversion theorem and a Calderón reproducing formula. As applications on the ( k ,  a )-generalized wavelet transform, we give some applications of the theory of reproducing kernels to the Tikhonov regularization on the generalized Sobolev spaces. Next, we study the generalized wavelet localization operators.
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ISSN:1662-9981
1662-999X
DOI:10.1007/s11868-019-00291-5