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The refinement equation corresponding to [Formula: see text] and [Formula: see text] is [Formula: see text] A solution [Formula: see text] if one exists, is called a refinable vector function or a vector scaling function of multiplicity\u00a0[Formula: see text] This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the [Formula: see text]-norm joint spectral radius of a finite set of finite matrices determined by the coefficients [Formula: see text]<\/jats:p>","DOI":"10.1142\/s0219691317500515","type":"journal-article","created":{"date-parts":[[2017,6,27]],"date-time":"2017-06-27T10:14:26Z","timestamp":1498558466000},"page":"1750051","source":"Crossref","is-referenced-by-count":0,"title":["Smoothness of refinable function vectors on \u211dn"],"prefix":"10.1142","volume":"15","author":[{"given":"Ramazan","family":"Tinaztepe","sequence":"first","affiliation":[{"name":"School of Supporting Studies, Imam Abdulrahman Bin Faisal University, (University of Dammam), Dammam 34212, KSA"}]},{"given":"Denise","family":"Jacobs","sequence":"additional","affiliation":[{"name":"Ellicott City, MD 21041, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2873-3436","authenticated-orcid":false,"given":"Christopher","family":"Heil","sequence":"additional","affiliation":[{"name":"School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA"}]}],"member":"219","published-online":{"date-parts":[[2017,8,28]]},"reference":[{"key":"S0219691317500515BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2006.05.008"},{"key":"S0219691317500515BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90267-E"},{"key":"S0219691317500515BIB003","doi-asserted-by":"publisher","DOI":"10.1006\/jath.1997.3211"},{"issue":"807","key":"S0219691317500515BIB004","first-page":"1","volume":"170","author":"Cabrelli C.","year":"2004","journal-title":"Mem. 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