Approximation of a function \(f\) of generalized Lipschitz class by its extended Legendre wavelet series
type: Item
instance of: scholarly article
title: Approximation of a function \(f\) of generalized Lipschitz class by its extended Legendre wavelet series
zbMATH Open document ID: 1412.42092
author: Shyam Lal
author: Priya Kumari
published in: International Journal of Applied and Computational Mathematics
publication date: 2018-12-03T00:00:00Z
review text: The authors propose two generalizations of the class of \(\alpha\)-Lipschitz functions on $[0,1)$. For some $\alpha \in (0,1]$, these classes are $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ and $\operatorname{Lip}\xi [0,1]$, where $f\in\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ if \[ \left\vert f(x+t)-f(x)\right\vert =O\left( s(t)\left\vert t\right\vert ^{\alpha}\right),\quad 0<\alpha \leq 1,\quad x,x+t\in [0,1], \] and \(f\in\operatorname{Lip}\xi [0,1]\) if \[ \vert f(x)-f(y)\vert =O(\xi\vert x-y\vert),\quad x,y\in [0,1]. \] Here, $s$ is positive, increasing, such that $s(t)\vert t\vert ^{\alpha}\rightarrow 0$ as $t\rightarrow 0^{+}$ and, $\xi $ is increasing, such that $\xi (t)\rightarrow 0$ as $t\rightarrow 0^{+}$. Considering the generalized Legendre wavelet expansions for functions belonging to the classes $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ and \(\operatorname{Lip}\xi [0,1]\), several estimations are obtained for these functions with respect to the \(L_{2}\) norm including some corollaries for the $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ classes.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 6987687
zbMATH Keywords: \(\operatorname{Lip}\xi[0,1)\) class
zbMATH Keywords: \(\operatorname{Lip}_\alpha ^{(s)}[0,1)\) class
zbMATH Keywords: Legendre wavelet
zbMATH Keywords: extended Legendre wavelet
zbMATH Keywords: extended Legendre expansion
zbMATH Keywords: extended Legendre approximation
reviewed by: Lucian Coroianu
MaRDI profile type: Publication
full work available at URL: s40819-018-0577-8
OpenAlex ID: W2901172806
cites work: Q3254471
cites work: Q4791729
cites work: Legendre wavelet method for numerical solutions of partial differential equations
cites work: Q4215356
cites work: Q4608533
cites work: Q4359331
cites work: Collocation method for the natural boundary integral equation
DOI: 10.1007/S40819-018-0577-8
description: scientific article; zbMATH DE number 6987687