On \(q\)-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet
type: Item
instance of: scholarly article
title: On \(q\)-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet
zbMATH Open document ID: 1385.42032
published in: Applied and Computational Harmonic Analysis
publication date: 2018-01-12T00:00:00Z
review text: The authors study in great detail a number of so called \(q\)-advanced special functions that converge to their classical counterparts as \(q\to 1+\) and the objective is to obtain functions with desirable wavelet properties and study their properties. The \(q\)-advanced spherical Bessel functions \(j_n(q;t)\) are defined by \[ j_n(q;t)=(-1)^n\left (\frac 1t \frac {d}{dt}\right )^n \frac {{}_qSin(t)}{t}, \] where \(n\in \mathbb N_0\), \(q>1\), and \[ {}_qSin(t)=\text{sign}(t)\frac{\sum_{M=-\infty}^\infty (-1)^Me^{-q^M|t|}q^{-M(M-1)}}{\sum_{M=-\infty}^\infty (-1)^Mq^{-M^2}}, \] (with a similar definition for \({}_qCos(t)\)). The functions \(j_n(q;t)\) are shown to belong to the Schwartz class and satisfy the multiplicative advanced differential equation \[ j_n''(q;t)+ \frac 2t j_n'(q;t)- \frac {n(n+1)}{t^2} j_n(q;t)= -q^{n+3}j_n(q;qt), \] and each of the odd-order functions \(j_{2k+1}(q;t)\) is a mother wavelet generating a frame for \(\mathcal L^2(\mathbb R)\). Other special functions studied are the inverse Fourier transforms \(\mathcal F^{-1}[j_n(q;t)](\omega)\) and \(\mathcal F^{-1}[\sin(t)j_n(q;t)](\omega)\). The former leads to \(q\)-advanced Legendre polynomials and from the latter one gets a \(\mathcal C^\infty\)-perturbation of the Haar wavelet that gives an almost orthogonal frame for \(\mathcal L^2(\mathbb R)\).
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 33E99
zbMATH DE Number: 6825639
zbMATH Keywords: \(q\)-advanced spherical Bessel functions
zbMATH Keywords: \(q\)-advanced Haar wavelet
zbMATH Keywords: \(q\)-Legendre polynomials
zbMATH Keywords: advanced differential equation
zbMATH Keywords: wavelet frame
zbMATH Keywords: Jacobi theta function
zbMATH Keywords: \(q\)-advanced sine
zbMATH Keywords: \(q\)-advanced cosine
zbMATH Keywords: smooth approximations to the Haar wavelet
author: D. W. Pravica
author: N. Randriampiry
author: M. J. Spurr
MaRDI profile type: Publication
full work available at URL: j.acha.2016.05.003
OpenAlex ID: W2404873307
cites work: Q5558293
cites work: Bessel sequences and affine frames
cites work: An introduction to frames and Riesz bases
cites work: Ten Lectures on Wavelets
cites work: Asymptotic behavior of linear advanced differential equations
cites work: On the stability of frames and Riesz bases
cites work: Bessel wavelets and the Galerkin analysis of the Bessel operator
cites work: Q4040884
cites work: Perturbations of the Haar wavelet
cites work: On \(q\)-asymptotics for linear \(q\)-difference-differential equations with Fuchsian and irregular singularities
cites work: On \(q\)-Gevrey asymptotics for singularly perturbed \(q\)-difference-differential problems with an irregular singularity
cites work: On parametric Gevrey asymptotics for singularly perturbed partial differential equations with delays
cites work: On complex singularity analysis for linear partial \(q\)-difference-differential equations using nonlinear differential equations
cites work: Q3135829
cites work: Q4338833
cites work: Applications of an advanced differential equation in the study of wavelets
cites work: Theta function identities in the study of wavelets satisfying advanced differential equations
cites work: Reproducing kernel bounds for an advanced wavelet frame via the theta function
cites work: \(q\)-advanced models for tsunami and rogue waves
cites work: Smooth wavelet approximations of truncated Legendre polynomials via the Jacobi theta function
cites work: Q4066072
cites work: Q3955832
cites work: About Jackson \(q\)-Bessel functions
DOI: 10.1016/J.ACHA.2016.05.003
description: scientific article; zbMATH DE number 6825639