(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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Vydané v:	Journal of pseudo-differential operators and applications Ročník 11; číslo 1; s. 55 - 92
Hlavný autor:	Mejjaoli, Hatem
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.03.2020 Springer Nature B.V
Predmet:	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
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Abstract	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.
AbstractList	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. 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.
Author	Mejjaoli, Hatem
Author_xml	– sequence: 1 givenname: Hatem surname: Mejjaoli fullname: Mejjaoli, Hatem email: hmejjaoli@gmail.com organization: Department of Mathematics, College of Sciences, Taibah University
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Cites_doi	10.1112/S0010437X11007445 10.1515/9781400882427 10.1016/j.jmaa.2006.04.092 10.1080/10652469.2011.647015 10.1090/S0002-9947-1956-0082586-0 10.1007/s00233-014-9617-9 10.1090/S0002-9947-2012-05608-X 10.1142/9789814503747_0003 10.1007/978-3-0348-8217-0 10.1090/conm/138/1199124 10.1002/mma.2679 10.1090/S0002-9947-1989-0951883-8 10.1364/JOSAA.14.001467 10.1088/1751-8113/44/35/355205 10.1016/j.jmaa.2017.12.018 10.1007/s00041-005-4079-9 10.1093/imrn/rnv398 10.1007/s11868-018-0260-1 10.1080/10652469.2013.799467 10.1137/0515056 10.1142/S0129167X16500191 10.1215/S0012-7094-99-09813-7 10.1090/pspum/048/974332 10.1093/oso/9780198534815.001.0001 10.1137/1.9781611970104 10.1016/0016-7142(84)90025-5 10.1090/S0065-9266-2011-00592-7
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Snippet	We introduce the notion of the ( k , a )-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl... We introduce the notion of the (k, a)-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl...
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SubjectTerms	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
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Title	(k,a)-generalized wavelet transform and applications
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