{"id":"https://openalex.org/W2132121843","doi":"https://doi.org/10.1109/tassp.1987.1165145","title":"On the rate of growth of condition numbers for convolution matrices","display_name":"On the rate of growth of condition numbers for convolution matrices","publication_year":1987,"publication_date":"1987-04-01","ids":{"openalex":"https://openalex.org/W2132121843","doi":"https://doi.org/10.1109/tassp.1987.1165145","mag":"2132121843"},"language":"en","primary_location":{"id":"doi:10.1109/tassp.1987.1165145","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tassp.1987.1165145","pdf_url":null,"source":{"id":"https://openalex.org/S98930721","display_name":"IEEE Transactions on Acoustics Speech and Signal Processing","issn_l":"0096-3518","issn":["0096-3518"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Acoustics, Speech, and Signal Processing","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/A5048678228","display_name":"F. Milinazzo","orcid":"https://orcid.org/0000-0002-6338-1547"},"institutions":[{"id":"https://openalex.org/I65595360","display_name":"Royal Roads University","ror":"https://ror.org/05w4ste42","country_code":"CA","type":"education","lineage":["https://openalex.org/I65595360"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"F. Milinazzo","raw_affiliation_strings":["Mathematics Department, Royal Roads Millitary College, Victoria, BC, Canada"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, Royal Roads Millitary College, Victoria, BC, Canada","institution_ids":["https://openalex.org/I65595360"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063296044","display_name":"Cedric A. Zala","orcid":null},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"C. Zala","raw_affiliation_strings":["Computer Science Department, University of Victoria, Victoria, BC, Canada"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, University of Victoria, Victoria, BC, Canada","institution_ids":["https://openalex.org/I212119943"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061073305","display_name":"I. Barrodale","orcid":null},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"I. Barrodale","raw_affiliation_strings":["Computer Science Department, University of Victoria, Victoria, BC, Canada"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, University of Victoria, Victoria, BC, Canada","institution_ids":["https://openalex.org/I212119943"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5048678228"],"corresponding_institution_ids":["https://openalex.org/I65595360"],"apc_list":null,"apc_paid":null,"fwci":1.0314,"has_fulltext":false,"cited_by_count":23,"citation_normalized_percentile":{"value":0.76924405,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"35","issue":"4","first_page":"471","last_page":"475"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9904000163078308,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9904000163078308,"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"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9732000231742859,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12510","display_name":"Magneto-Optical Properties and Applications","score":0.972100019454956,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6389287710189819},{"id":"https://openalex.org/keywords/convolution","display_name":"Convolution (computer science)","score":0.6216781735420227},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.6199549436569214},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.5993592143058777},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5796230435371399},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5502524375915527},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.5296021699905396},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.5030490756034851},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.48101526498794556},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4733167886734009},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43390369415283203},{"id":"https://openalex.org/keywords/deconvolution","display_name":"Deconvolution","score":0.41235843300819397},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3634606599807739},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.17472344636917114},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16383293271064758},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.14965561032295227},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.147962749004364},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09737721085548401},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.0868312418460846}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6389287710189819},{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.6216781735420227},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.6199549436569214},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.5993592143058777},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5796230435371399},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5502524375915527},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.5296021699905396},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.5030490756034851},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.48101526498794556},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4733167886734009},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43390369415283203},{"id":"https://openalex.org/C174576160","wikidata":"https://www.wikidata.org/wiki/Q1183700","display_name":"Deconvolution","level":2,"score":0.41235843300819397},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3634606599807739},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.17472344636917114},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16383293271064758},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.14965561032295227},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.147962749004364},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09737721085548401},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0868312418460846},{"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/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tassp.1987.1165145","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tassp.1987.1165145","pdf_url":null,"source":{"id":"https://openalex.org/S98930721","display_name":"IEEE Transactions on Acoustics Speech and Signal Processing","issn_l":"0096-3518","issn":["0096-3518"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Acoustics, Speech, and Signal Processing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W833302436","https://openalex.org/W1995240069","https://openalex.org/W2009410052","https://openalex.org/W2010355946","https://openalex.org/W2045463928","https://openalex.org/W2312497442","https://openalex.org/W2330746718","https://openalex.org/W4388534025"],"related_works":["https://openalex.org/W2369061952","https://openalex.org/W2367122702","https://openalex.org/W1601492201","https://openalex.org/W4231036715","https://openalex.org/W4319586039","https://openalex.org/W2132989621","https://openalex.org/W2383495548","https://openalex.org/W2370645350","https://openalex.org/W2015447694","https://openalex.org/W1969590113"],"abstract_inverted_index":{"When":[0],"analyzing":[1],"linear":[2],"systems":[3],"of":[4,10,17,53,68,98,105,109,153,161,165,167,171,179,184,188,195,214,221,224],"equations,":[5],"the":[6,14,18,51,54,58,69,90,96,99,102,106,132,154,163,168,180,185,211,219,222,225],"most":[7],"important":[8],"indicator":[9],"potential":[11],"instability":[12],"is":[13,93,113,197,210,218],"condition":[15,48,70,91,134,169],"number":[16,49,71,92,135,170],"matrix.":[19],"For":[20,57],"a":[21,27,120,149,159,174],"convolution":[22,60,84],"matrix":[23],"W":[24,137,172,215],"formed":[25],"from":[26],"series":[28,117,176],"w":[29,118,189],"(where":[30],"<tex":[31],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[32,205],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">W_{ij}":[33],"=":[34,44],"w_{i-j+1},":[35],"1":[36,40],"\\leq":[37,41],"i-j":[38],"+":[39],"k,":[42],"W_{ij}":[43],"0</tex>":[45],"otherwise),":[46],"this":[47,145],"defines":[50],"stability":[52],"deconvolution":[55],"process.":[56],"larger":[59],"matrices":[61],"commonly":[62],"encountered":[63],"in":[64,123,139],"practice,":[65],"direct":[66],"computation":[67],"(e.g.,":[72],"by":[73,95,177],"singular":[74],"value":[75,122],"decomposition)":[76],"would":[77],"be":[78,141,200],"extremely":[79],"time":[80],"consuming.":[81],"However,":[82],"for":[83,89,115,128,136,173],"matrices,":[85],"an":[86],"upper":[87,155],"bound":[88,112,156],"defined":[94],"ratio":[97],"maximum":[100],"to":[101,199,202],"minimum":[103],"values":[104],"amplitude":[107,125,186,226],"spectrum":[108,187],"w.":[110],"This":[111],"infinite":[114],"any":[116],"with":[119],"zero":[121,223],"its":[124,191],"spectrum;":[126],"although":[127],"certain":[129],"such":[130],"series,":[131],"actual":[133],"may":[138],"fact":[140],"relatively":[142],"small.":[143],"In":[144],"paper":[146],"we":[147],"give":[148],"new":[150],"simple":[151],"derivation":[152],"and":[157,216],"present":[158],"means":[160,178],"defining":[162],"rate":[164,194],"growth":[166,196],"band-limited":[175],"higher":[181],"order":[182,220],"derivatives":[183],"at":[190],"zeros.":[192],"The":[193],"shown":[198],"proportional":[201],"m":[203,209],"<sup":[204],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">p</sup>":[206],",":[207],"where":[208],"column":[212],"dimension":[213],"p":[217],"spectrum.":[227]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}