Pseudodifferential operators on Adele rings and wavelet bases
type: Item
instance of: scholarly article
title: Pseudodifferential operators on Adele rings and wavelet bases
zbMATH Open document ID: 1263.47062
DOI: 10.1134/S1064562412030167
author: Andrei Khrennikov
author: A. V. Kosyak
author: V. M. Shelkovich
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Let \(\mathbb Q_p\) be the field of \(p\)-adic numbers and \(\mathbb Z_p=\{x\in\mathbb Q_p: |x|_p\leq1\}\) be the subring of integers in \(\mathbb Q_p\). Suppose that there exists a prime \(P = P(x)\) such that \(x_p\in\mathbb Z_p\) for all \(p>P\). The authors define the adele ring \(\mathbb A\) as the set of infinite sequences \(x'= (x_2,\dots, x_p,\dots)\), where \(x_p\in \mathbb Q_p\) for \(p = 2,3,\dots\). They construct wavelet bases in \(L^2(\mathbb Q_p)\) for \(p\in V_\mathbb Q\) and in \(L^2(\mathbb A)\). Then they introduce Lizorkin spaces \(\Phi(\mathbb A)\) on adeles and wavelets, leading to Theorem~1. A~class of pseudodifferential operators is defined on \(\Phi(\mathbb A)\), leading to Theorem~2. A~fractional adelic operator is introduced and eigenfunctions of pseudodifferential operators are considered in Theorem~3.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 6106461
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: adele ring
zbMATH Keywords: wavelet basis
MaRDI profile type: Publication
cites work: Q5306363
cites work: \(p\)-adic orthogonal wavelet bases
cites work: Wavelet bases on adele rings
cites work: Q5431578
cites work: \(p\)-adic Haar multiresolution analysis and pseudo-differential operators
cites work: Q4726489
cites work: Q4003475
description: scientific article; zbMATH DE number 6106461