{"id":"https://openalex.org/W2001560283","doi":"https://doi.org/10.1137/s089547989732383x","title":"Smoothness of Multiple Refinable Functions and Multiple Wavelets","display_name":"Smoothness of Multiple Refinable Functions and Multiple Wavelets","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W2001560283","doi":"https://doi.org/10.1137/s089547989732383x","mag":"2001560283"},"language":"en","primary_location":{"id":"doi:10.1137/s089547989732383x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s089547989732383x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://scholars.cityu.edu.hk/en/publications/smoothness-of-multiple-refinable-functions-and-multiple-wavelets","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112272485","display_name":"Rong-Qing Jia","orcid":null},"institutions":[{"id":"https://openalex.org/I154425047","display_name":"University of Alberta","ror":"https://ror.org/0160cpw27","country_code":"CA","type":"education","lineage":["https://openalex.org/I154425047"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Rong-Qing Jia","raw_affiliation_strings":["Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1","institution_ids":["https://openalex.org/I154425047"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009736660","display_name":"S. D. Riemenschneider","orcid":null},"institutions":[{"id":"https://openalex.org/I154425047","display_name":"University of Alberta","ror":"https://ror.org/0160cpw27","country_code":"CA","type":"education","lineage":["https://openalex.org/I154425047"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Sherman D. Riemenschneider","raw_affiliation_strings":["Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1","institution_ids":["https://openalex.org/I154425047"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006802589","display_name":"Ding\u2010Xuan Zhou","orcid":"https://orcid.org/0000-0003-0224-9216"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Ding-Xuan Zhou","raw_affiliation_strings":["Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I168719708"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5112272485"],"corresponding_institution_ids":["https://openalex.org/I154425047"],"apc_list":null,"apc_paid":null,"fwci":9.6129,"has_fulltext":false,"cited_by_count":81,"citation_normalized_percentile":{"value":0.98560886,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"21","issue":"1","first_page":"1","last_page":"28"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9948999881744385,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7653940916061401},{"id":"https://openalex.org/keywords/spectral-radius","display_name":"Spectral radius","score":0.6529665589332581},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6235955953598022},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.5996959805488586},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.5972886085510254},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.49493688344955444},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4916535019874573},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.48337382078170776},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4363196790218353},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4223138391971588},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.4209122061729431},{"id":"https://openalex.org/keywords/invariant-subspace","display_name":"Invariant subspace","score":0.41840052604675293},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31514453887939453},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22581037878990173},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.18811967968940735},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14818677306175232}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7653940916061401},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.6529665589332581},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6235955953598022},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.5996959805488586},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.5972886085510254},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.49493688344955444},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4916535019874573},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.48337382078170776},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4363196790218353},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4223138391971588},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.4209122061729431},{"id":"https://openalex.org/C2777059694","wikidata":"https://www.wikidata.org/wiki/Q2706744","display_name":"Invariant subspace","level":3,"score":0.41840052604675293},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31514453887939453},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22581037878990173},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.18811967968940735},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14818677306175232},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/s089547989732383x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s089547989732383x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/smoothness-of-multiple-refinable-functions-and-multiple-wavelets","pdf_url":null,"source":{"id":"https://openalex.org/S7407055387","display_name":"CityU Scholars","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Jia, R-Q, Riemenschneider, S D & Zhou, D-X 1999, 'Smoothness of multiple refinable functions and multiple wavelets', SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 1, pp. 1-28. https://doi.org/10.1137/S089547989732383X","raw_type":"article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.124.975","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.975","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ualberta.ca/~rjia/Paper96-00/JRZc.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.49.4632","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.4632","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://approx.math.ualberta.ca/pub/Riemen/jrz.smrfmw.a4.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.54.7851","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.7851","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://approx.math.ualberta.ca/pub/Riemen/jrz.smrfmw.970617.ps.gz","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/227e7a8c-4711-4aaa-ac65-5cbf1bcbc406","is_oa":true,"landing_page_url":"https://hdl.handle.net/2031/227e7a8c-4711-4aaa-ac65-5cbf1bcbc406","pdf_url":null,"source":{"id":"https://openalex.org/S7407055387","display_name":"CityU Scholars","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Jia, R-Q, Riemenschneider, S D & Zhou, D-X 1999, 'Smoothness of multiple refinable functions and multiple wavelets', SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 1, pp. 1-28. https://doi.org/10.1137/S089547989732383X","raw_type":"article"}],"best_oa_location":{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/smoothness-of-multiple-refinable-functions-and-multiple-wavelets","pdf_url":null,"source":{"id":"https://openalex.org/S7407055387","display_name":"CityU Scholars","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Jia, R-Q, Riemenschneider, S D & Zhou, D-X 1999, 'Smoothness of multiple refinable functions and multiple wavelets', SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 1, pp. 1-28. https://doi.org/10.1137/S089547989732383X","raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W146269660","https://openalex.org/W1774981766","https://openalex.org/W1964715865","https://openalex.org/W1972706755","https://openalex.org/W1978437745","https://openalex.org/W1979032637","https://openalex.org/W1980695038","https://openalex.org/W1981425820","https://openalex.org/W1987996534","https://openalex.org/W1995668896","https://openalex.org/W1998722818","https://openalex.org/W2003989039","https://openalex.org/W2004766860","https://openalex.org/W2006434287","https://openalex.org/W2022993606","https://openalex.org/W2026580946","https://openalex.org/W2030934791","https://openalex.org/W2032326995","https://openalex.org/W2032753235","https://openalex.org/W2051656206","https://openalex.org/W2056166036","https://openalex.org/W2060971311","https://openalex.org/W2079074323","https://openalex.org/W2081635712","https://openalex.org/W2090408827","https://openalex.org/W2090953263","https://openalex.org/W2109411320","https://openalex.org/W2113712067","https://openalex.org/W2116124511","https://openalex.org/W4254836152","https://openalex.org/W4388321579"],"related_works":["https://openalex.org/W4235906205","https://openalex.org/W4301478900","https://openalex.org/W2963446266","https://openalex.org/W2333831800","https://openalex.org/W2003960350","https://openalex.org/W102084567","https://openalex.org/W2948021207","https://openalex.org/W2118409924","https://openalex.org/W4221145148","https://openalex.org/W2978618494"],"abstract_inverted_index":{"We":[0,46,214],"consider":[1],"the":[2,14,21,43,48,70,75,81,108,129,136,161,173,182,186,191,196,207,220,230],"smoothness":[3,60,175,209,221],"of":[4,6,9,23,37,61,80,131,148,157,164,181,185,203,223,235],"solutions":[5],"a":[7,31,33,62,99,200],"system":[8],"refinement":[10,44,192],"equations":[11],"written":[12],"in":[13,27,179],"form":[15],"$$\\phi":[16],"=":[17,86,92,171],"\\sum\\nolimits_{\\ga\\in\\ZZ}":[18],"a(\\ga)\\phi({2\\,\\cdot}-\\ga),$$":[19],"where":[20,206],"vector":[22],"functions":[24],"$\\phi=(\\phi_1,\\ldots,\\phi_r)^T$":[25],"is":[26,32,67,139,167,176],"$(L_p(\\mbox{\\smallBbb":[28],"R}))^r$":[29],"and":[30,135,232],"finitely":[34],"supported":[35],"sequence":[36],"r":[38,40],"X":[39],"matrices":[41,83],"called":[42],"mask.":[45,193],"use":[47,156],"generalized":[49],"Lipschitz":[50],"space":[51],"$\\hbox{Lip}^*(\\nu,":[52],"L_p(\\mbox{\\smallBbb":[53],"R}))$,":[54],"$\\nu":[55],">":[56],"0$,":[57],"to":[58,68,74,98,219],"measure":[59],"given":[63,89,178,212],"function.":[64],"Our":[65],"method":[66,166],"relate":[69],"optimal":[71,174,208],"smoothness,":[72],"$\\nu_p(\\phi)$,":[73],"p-norm":[76,109],"joint":[77,110],"spectral":[78,111,183],"radius":[79,112,184],"block":[82],"$A_\\gep$,":[84],"$\\gep":[85],"0,":[87],"1$,":[88],"by":[90,113,141,145],"$A_\\gep":[91],"(":[93],"a(\\gep+2\\ga-\\gb)":[94],")_{\\ga,\\gb}$,":[95],"when":[96,128],"restricted":[97],"certain":[100,142],"finite":[101],"dimensional":[102],"common":[103],"invariant":[104,137],"subspace":[105,138],"V.":[106],"Denoting":[107],"$\\rho_p(A_0|_V,A_1|_V)$,":[114],"we":[115,198],"show":[116],"that":[117],"$\\nu_p(\\phi)":[118],"\\ge":[119],"1/p":[120],"-":[121],"\\log_2":[122],"\\rho_p(A_0|_V,":[123],"A_1|_V)":[124],"$":[125],"with":[126,190],"equality":[127],"shifts":[130],"$\\phi_1,\\ldots,\\phi_r$":[132],"are":[133],"stable":[134],"generated":[140],"vectors":[143],"induced":[144],"difference":[146],"operators":[147],"sufficiently":[149],"high":[150],"order.":[151],"This":[152],"allows":[153],"an":[154],"effective":[155],"matrix":[158,188],"theory.":[159],"Also":[160],"computational":[162],"implementation":[163],"our":[165,217,236],"simple.":[168],"When":[169],"p":[170],"2,":[172],"also":[177,215],"terms":[180],"transition":[187],"associated":[189],"To":[194],"illustrate":[195],"theory,":[197],"give":[199],"detailed":[201],"analysis":[202,222],"two":[204],"examples":[205,227],"can":[210],"be":[211],"explicitly.":[213],"apply":[216],"methods":[218],"multiple":[224],"wavelets.":[225],"These":[226],"clearly":[228],"demonstrate":[229],"applicability":[231],"practical":[233],"power":[234],"approach.":[237]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":4},{"year":2013,"cited_by_count":2}],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2025-10-10T00:00:00"}
