{"id":"https://openalex.org/W1966975678","doi":"https://doi.org/10.1137/s0895479801397858","title":"Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets","display_name":"Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W1966975678","doi":"https://doi.org/10.1137/s0895479801397858","mag":"1966975678"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479801397858","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479801397858","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"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"],"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/A5027322245","display_name":"Qingtang Jiang","orcid":"https://orcid.org/0000-0002-0173-9988"},"institutions":[{"id":"https://openalex.org/I12097938","display_name":"West Virginia University","ror":"https://ror.org/011vxgd24","country_code":"US","type":"education","lineage":["https://openalex.org/I12097938"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Qingtang Jiang","raw_affiliation_strings":["Department of Mathematics, West Virginia University, Morgantown, WV 26506, U.S.A"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, West Virginia University, Morgantown, WV 26506, U.S.A","institution_ids":["https://openalex.org/I12097938"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5112272485"],"corresponding_institution_ids":["https://openalex.org/I154425047"],"apc_list":null,"apc_paid":null,"fwci":10.5164,"has_fulltext":false,"cited_by_count":88,"citation_normalized_percentile":{"value":0.98662497,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"24","issue":"4","first_page":"1071","last_page":"1109"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9993000030517578,"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":0.9993000030517578,"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":0.9991000294685364,"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/combinatorics","display_name":"Combinatorics","score":0.7123631238937378},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6724108457565308},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.5617372393608093},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.5100861191749573},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.45352041721343994},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.448302686214447},{"id":"https://openalex.org/keywords/expansive","display_name":"Expansive","score":0.44015902280807495},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4283900856971741},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.34954771399497986},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2831750810146332},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26594454050064087}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7123631238937378},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6724108457565308},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.5617372393608093},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.5100861191749573},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.45352041721343994},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.448302686214447},{"id":"https://openalex.org/C2780502288","wikidata":"https://www.wikidata.org/wiki/Q28838156","display_name":"Expansive","level":3,"score":0.44015902280807495},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4283900856971741},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.34954771399497986},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2831750810146332},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26594454050064087},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C30407753","wikidata":"https://www.wikidata.org/wiki/Q186191","display_name":"Compressive strength","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/s0895479801397858","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479801397858","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.111.970","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.970","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/Paper01-05/JJb.pdf","raw_type":"text"}],"best_oa_location":null,"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":39,"referenced_works":["https://openalex.org/W146269660","https://openalex.org/W1277493121","https://openalex.org/W1487638357","https://openalex.org/W1488877410","https://openalex.org/W1492181480","https://openalex.org/W1717630816","https://openalex.org/W1774981766","https://openalex.org/W1964715865","https://openalex.org/W1971998705","https://openalex.org/W1973578161","https://openalex.org/W1984093439","https://openalex.org/W1986784032","https://openalex.org/W1986869180","https://openalex.org/W1987996534","https://openalex.org/W1995668896","https://openalex.org/W1998570043","https://openalex.org/W2001560283","https://openalex.org/W2004766860","https://openalex.org/W2008311911","https://openalex.org/W2019113134","https://openalex.org/W2030772655","https://openalex.org/W2030934791","https://openalex.org/W2032753235","https://openalex.org/W2036959719","https://openalex.org/W2046153167","https://openalex.org/W2048247182","https://openalex.org/W2056166036","https://openalex.org/W2061160109","https://openalex.org/W2090055402","https://openalex.org/W2092896986","https://openalex.org/W2128978199","https://openalex.org/W2144471752","https://openalex.org/W2147375939","https://openalex.org/W2157565801","https://openalex.org/W2169447070","https://openalex.org/W2341241031","https://openalex.org/W4205354563","https://openalex.org/W4254836152","https://openalex.org/W4297562995"],"related_works":["https://openalex.org/W1973865997","https://openalex.org/W4253570894","https://openalex.org/W2604017320","https://openalex.org/W4288323582","https://openalex.org/W1978042415","https://openalex.org/W1535410608","https://openalex.org/W1559102282","https://openalex.org/W2025296150","https://openalex.org/W2281786397","https://openalex.org/W1967448091"],"abstract_inverted_index":{"This":[0,245],"paper":[1,203],"investigates":[2],"spectral":[3],"properties":[4],"of":[5,22,24,29,39,62,154,165,201,253],"the":[6,20,25,67,144,150,237,265],"transition":[7],"operator":[8],"associated":[9],"to":[10,19,101,263],"a":[11,91,110],"multivariate":[12],"vector":[13,28,38],"refinement":[14],"equation":[15],"and":[16,108,132,161,172,234,256],"their":[17],"applications":[18],"study":[21],"smoothness":[23,61],"corresponding":[26],"refinable":[27],"functions.":[30],"Let":[31,126],"$\\Phi=(\\phi_1,\\ldots,\\phi_r)^T$":[32],"be":[33],"an":[34,56,250],"$r":[35],"\\times":[36],"1$":[37],"compactly":[40],"supported":[41],"functions":[42],"in":[43],"$L_2(\\Rs)$":[44],"satisfying":[45],"$\\,\\Phi":[46],"=":[47,212],"\\sum_{\\ga\\in\\Zs}":[48,129],"a(\\ga)":[49,113],"\\Phi({M\\kern":[50],".1em":[51],"\\cdot}-\\ga)$,":[52],"where":[53,141,218],"M":[54,98],"is":[55,64,92,99,116,159,167,208,236,247],"expansive":[57],"integer":[58],"matrix.":[59],"The":[60,198],"$\\Phi$":[63,158,207],"measured":[65],"by":[66,157],"Sobolev":[68],"critical":[69],"exponent":[70],"$\\gl":[71],"(\\Phi)":[72],":=":[73,111,122,128,134,182,220],"\\sup":[74],"\\bigl\\{":[75],"\\gl:":[76],"\\int_{\\Rs}":[77],"|\\hat":[78],"\\phi_j(\\xi)|^2(1+|\\xi|^\\gl)^2\\,d\\xi":[79],"<":[80,189,195],"\\infty,":[81],"\\,":[82],"1":[83],"\\le":[84],"j\\le":[85,177],"r":[86],"\\bigr\\}$.":[87],"Suppose":[88,148],"%the":[89,96],"mask":[90],"finitely":[93],"supported,":[94],"i.e.,":[95],"set":[97,238],"similar":[100],"%a":[102],"diagonal":[103],"matrix":[104,257],"$\\hbox{diag}(\\gs_1,\\ldots,\\gs_s)$":[105],"with":[106,170],"$|\\gs_1|=\\cdots=|\\gs_s|$":[107],"$\\supp":[109],"\\{\\ga\\in\\Zs:":[112],"\\ne":[114,174],"0\\}$":[115],"finite.":[117],"For":[118],"$\\mu=(\\mu_1,\\ldots,\\mu_s)\\in\\NN_0^s$,":[119],"define":[120],"$\\gs^{-\\mu}":[121],"\\gs_1^{-\\mu_1}":[123],"\\cdots":[124],"\\gs_s^{-\\mu_s}$.":[125],"$A":[127],"a(\\ga)/|\\det":[130],"M|$":[131],"$b(\\ga)":[133],"\\sum_{\\gb\\in\\Zs}":[135],"\\overline{a(\\gb)}":[136],"\\otimes":[137],"a(\\ga+\\gb)/|\\det":[138],"M|$,":[139],"$\\ga\\in\\Zs$,":[140],"$\\,\\otimes\\,$":[142],"denotes":[143],"(right)":[145],"Kronecker":[146],"product.":[147],"that":[149,205],"highest":[151],"total":[152],"degree":[153],"polynomials":[155],"reproduced":[156],"$k-1$":[160],"$\\spec(A)$":[162],"(the":[163],"spectrum":[164],"A)":[166],"$\\{\\eta_1,\\eta_2,":[168],"\\ldots,\\eta_r\\}$":[169],"$\\eta_1=1$":[171],"$\\eta_j":[173],"1,":[175],"2\\le":[176],"r$.":[178],"Set":[179],"$$":[180,197,233],"E_k":[181,231],"\\{":[183],"\\eta_j":[184],"\\overline{\\gs^{-\\mu}},":[185],"\\overline{\\eta_j}":[186],"\\gs^{-\\mu}:":[187],"|\\mu|":[188,194],"k,":[190],"j=2,\\ldots,r\\}":[191],"\\cup":[192],"\\{\\gs^{-\\mu}:":[193],"2k\\}.":[196],"main":[199],"result":[200,246],"this":[202],"asserts":[204],"if":[206],"stable,":[209],"then":[210],"$\\gl(\\Phi)":[211],"-":[213],"(\\log_{|\\det":[214],"M|}":[215],"\\rho_k)\\,":[216],"s/2$,":[217],"$$\\rho_k":[219],"\\max\\Big\\{\\,":[221],"|\\nu|:":[222],"\\nu":[223],"\\in":[224],"\\hbox{\\rm":[225],"spec}":[226],"\\bigl(b(M\\ga-\\gb)":[227],"\\bigr)_{\\ga,\\gb\\in":[228],"K}":[229],"\\setminus":[230],"\\,\\Big\\},":[232],"K":[235],"$\\Zs":[239],"\\cap":[240],"\\sum_{n=1}^\\infty":[241],"M^{-n}":[242],"(\\supp":[243],"b)$.":[244],"obtained":[248],"through":[249],"extensive":[251],"use":[252],"linear":[254],"algebra":[255],"theory.":[258,267],"Three":[259],"examples":[260],"are":[261],"provided":[262],"illustrate":[264],"general":[266]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
