Periodic wavelets on the \(p\)-adic Vilenkin group
type: Item
instance of: scholarly article
title: Periodic wavelets on the \(p\)-adic Vilenkin group
zbMATH Open document ID: 1256.42049
DOI: 10.1134/S2070046611040030
published in: \(p\)-Adic Numbers, Ultrametric Analysis, and Applications
publication date: 2012-11-13T00:00:00Z
review text: Let \((G,\oplus)\) be the locally compact \(p\)-adic Vilenkin group, \(U_0\) the unit ball, \(A\) the dilation operator and \(U_n=A^{-n}U_0\), \((w_l)_{l=0}^\infty\) Vilenkin functions. Using the modified Vilenkin-Dirichlet kernel \[ D_n^*(x):=\alpha+\sum_{k=1}^{N-l}w_k(x)+\alpha w_{N-1}(x), \] the author introduces the functions \[ \begin{multlined}\varphi_{n,k}(x)=\Phi_n(x\ominus x_{n,k}), \psi_{n,k}^{(j)}(x):=\\ \sum_{s=0}^{p-1}\varepsilon_p^{js}\varphi_{n+1,pk+s}(x), 0\leq k\leq N-1, 1\leq j\leq p-1, N=p^n,\end{multlined} \] where \(\Phi_n(x):=N^{-1}D_N^\alpha(x)\) and shows that \(\{\varphi_{n,k}\}_{k=0}^{N-1}\) and \(\{\psi_{n,k}^{(l)}\}_{k=0}^{N-1}\) are bases for the spaces \(V_n:=\text{span}\{1,w_1(x),\dots,w_{N-1}(x)\}\), \(W_n^{(j)}:=\text{span}\{w_{jN}(x),w_{jN+1}(x),\dots,w_{(j+1)N-1}(x)\}\) respectively. The corresponding fast algorithms for the decomposition and reconstruction are discussed.
reviewed by: S. F. Lukomskii
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 65T50
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 6105086
zbMATH Keywords: Dirichlet kernel
zbMATH Keywords: Cantor dyadic group
zbMATH Keywords: Vilenkin groups
zbMATH Keywords: periodic wavelets
zbMATH Keywords: multiresolution analusis
zbMATH Keywords: Walsh functions
zbMATH Keywords: discrete Vilenkin-Chrestenson transform
author: Yuri A. Farkov
MaRDI profile type: Publication
cites work: On trigonometric wavelets
cites work: Q3999467
cites work: Q3064733
cites work: Discrete wavelets and the Vilenkin-Chrestenson transform
cites work: ON BIORTHOGONAL WAVELETS RELATED TO THE WALSH FUNCTIONS
cites work: Algorithms for wavelet construction on Vilenkin groups
cites work: Q4898166
cites work: Wavelets based on orthogonal polynomials
cites work: Orthogonal Wavelets on the Cantor Dyadic Group
cites work: Q3997391
Recommended article: \(p\)-adic wavelets
Recommended article: \(p\)-adic wavelets and their applications
Recommended article: Frames of \(p\)-adic wavelets and orbits of the affine group
Recommended article: Q4995793
Recommended article: Wavelet theory as $ p$-adic spectral analysis
Recommended article: Approximation properties of systems of periodic wavelets on the Cantor group
Recommended article: On \(p\)-adic tight wavelet frames
Recommended article: Approximation properties of wavelets on p-adic fields
Recommended article: Wavelet frames on Vilenkin groups and their approximation properties
Recommended article: Q4986387
description: scientific article; zbMATH DE number 6105086