{"title":"Characterization of wavelets associated with $AB$-MRA on $L^2(\\mathbb R^n)$","authors":"O. Ahmad, M. Y. Bhat, N. Sheikh","doi":"10.52846/ami.v48i1.1446","DOIUrl":null,"url":null,"abstract":"A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2(\\mathbb R^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are matrices whose eigenvalues have magnitude one, while the members of $A$ are matrices expanding on a proper subspace of $\\mathbb R^n$. In this paper, we provide the characterization of composite wavelets based on results of affine and quasi affine frames. Furthermore all the composite wavelets associated with $AB$-MRA on $L^2(\\mathbb R^n)$ are also characterized.","PeriodicalId":43654,"journal":{"name":"Annals of the University of Craiova-Mathematics and Computer Science Series","volume":"22 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of the University of Craiova-Mathematics and Computer Science Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.52846/ami.v48i1.1446","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2(\mathbb R^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are matrices whose eigenvalues have magnitude one, while the members of $A$ are matrices expanding on a proper subspace of $\mathbb R^n$. In this paper, we provide the characterization of composite wavelets based on results of affine and quasi affine frames. Furthermore all the composite wavelets associated with $AB$-MRA on $L^2(\mathbb R^n)$ are also characterized.