Calderón's reproducing formula and uncertainty principle for the continuous wavelet transform associated with the \(q\)-Bessel operator
type: Item
instance of: scholarly article
title: Calderón's reproducing formula and uncertainty principle for the continuous wavelet transform associated with the \(q\)-Bessel operator
zbMATH Open document ID: 1402.42007
author: Bochra Nefzi
author: Kamel Brahim
published in: Journal of Pseudo-Differential Operators and Applications
publication date: 2018-09-19T00:00:00Z
review text: The paper falls into the scope of the \(q\)-harmonic analysis associated to the \(q\)-Bessel operator \[ \Delta_{q,\nu} f(x) =\frac{f(q^{-1}x)-(1+q^{2\nu}) f(x) + q^{2\nu} f(qx)}{x^2}, \qquad x\neq 0. \] This analysis includes the associated \(q\)-Bessel Fourier transform, the \(q\)-Bessel translation operator, and the qconvolution product. The authors introduce the concepts of \(q\)-wavelet and continuous \(q\)-wavelet transform and prove the corresponding analogue of Calderón's reproducing formula in the \(L^2\) setting. The uncertainty principle for these transforms is also studied.
reviewed by: Boris Rubin
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 33D05
zbMATH DE Number: 6939194
zbMATH Keywords: \(q\)-wavelet
zbMATH Keywords: continuous \(q\)-wavelet transform
zbMATH Keywords: \(q\)-Bessel operator
zbMATH Keywords: \(q\)-Bessel Fourier transform
zbMATH Keywords: Calderón's reproducing formula
zbMATH Keywords: uncertainty principle
zbMATH Keywords: Heisenberg-Pauli-Weyl inequality
MaRDI profile type: Publication
full work available at URL: s11868-017-0209-9
OpenAlex ID: W2611909878
cites work: Q5453762
cites work: Intermediate spaces and interpolation, the complex method
cites work: Ten Lectures on Wavelets
cites work: Q4791729
cites work: On the $q$-Bessel Fourier transform
cites work: Q5433822
cites work: Paley-Wiener theorem for the \(q\)-Bessel transform and associated \(q\)-sampling formula
cites work: On Hardy's inequality for symmetric integral transforms and analogous
cites work: Positivity of the generalized translation associated with the \(q\)-Hankel transform
cites work: The \textit{q-j}\(_\alpha\) Bessel function
cites work: Q3633356
cites work: Q4000004
cites work: The uncertainty principle: A mathematical survey
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Q4040884
cites work: Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
cites work: Pointwise analysis of Riemann's ``nondifferentiable'' function
cites work: A variation on uncertainty principles for the generalized \(q\)-Bessel Fourier transform
cites work: Quantum calculus
cites work: On the zeros of the Hahn-Exton \(q\)-Bessel function and associated \(q\)- Lommel polynomials
cites work: Q3135829
cites work: An uncertainty principle for Hankel transforms
cites work: An Addition Theorem and Some Product Formulas for the Hahn-Exton q-Bessel Functions
DOI: 10.1007/S11868-017-0209-9
description: scientific article; zbMATH DE number 6939194