Orthogonal polynomial wavelets
type: Item
instance of: scholarly article
title: Orthogonal polynomial wavelets
zbMATH Open document ID: 0998.42020
DOI: 10.1023/A:1015689418605
author: B. Fischer
author: Woula Themistoclakis
published in: Numerical Algorithms
publication date: 2002-07-08T00:00:00Z
review text: The authors construct orthogonal polynomial wavelets and extend some results of \textit{B. Fischer} and \textit{J. Prestin} [Math. Comput. 66, No. 220, 1593-1618 (1997; Zbl 0896.42020)]. Let \(P_j\) \((j= 0,1,\dots)\) be orthonormal polynomials on \([a,b]\) \((-\infty\leq a< b\leq\infty)\) with respect to a nonnegative weight function \(w\). Using the kernel polynomial \[ K_n(t,\xi)= \sum^n_{j=0} P_j(t) P_j(\xi), \] scaling functions and wavelets are introduced by \[ \varphi_{n,r}(t)= K_n(t, x^{(n+1)}_r)\qquad (r= 0,\dots, n), \] \[ \psi_{n,r}(t)= K_{2n}(t, z^{(n)}_r)- K_n(t, z^{(n)}_r)\qquad (r= 0,\dots, n-1), \] where \(x^{(n+1)}_0< x^{(n+1)}_1<\cdots< x^{(n+1)}_n\) and \(z^{(n)}_0< z^{(n)}_1<\cdots z^{(n)}_{n-1}\) are convenient parameters. The authors characterize those parameters \(z^{(n)}_r\) which lead to orthogonal wavelets. The corresponding decomposition and reconstruction schemes are sketched. Finally, the special case of Bernstein-Szegő weight function \(w\) is discussed.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1764839
zbMATH Keywords: orthogonal polynomial wavelets
zbMATH Keywords: orthonormal polynomials
zbMATH Keywords: kernel polynomial
zbMATH Keywords: orthogonal wavelets
zbMATH Keywords: Bernstein-Szegő weight function
MaRDI profile type: Publication
full work available at URL: a:1015689418605
OpenAlex ID: W2196060426
description: scientific article; zbMATH DE number 1764839