{"id":"https://openalex.org/W2016107283","doi":"https://doi.org/10.1137/070708937","title":"Uniform Approximation of Piecewise r-Smooth and Globally Continuous Functions","display_name":"Uniform Approximation of Piecewise r-Smooth and Globally Continuous Functions","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2016107283","doi":"https://doi.org/10.1137/070708937","mag":"2016107283"},"language":"en","primary_location":{"id":"doi:10.1137/070708937","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070708937","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","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/A5072564885","display_name":"Leszek Plaskota","orcid":"https://orcid.org/0000-0001-8704-0790"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Leszek Plaskota","raw_affiliation_strings":["leszekp@mimuw.edu.pl#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"leszekp@mimuw.edu.pl#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108372354","display_name":"Grzegorz W. Wasilkowski","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Grzegorz W. Wasilkowski","raw_affiliation_strings":["[email\u00a0protected]#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[email\u00a0protected]#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9546,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.74224453,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"47","issue":"1","first_page":"762","last_page":"785"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9872999787330627,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9805999994277954,"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":0.7017274498939514},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.573176383972168},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5047277212142944},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4682784080505371},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.45410168170928955},{"id":"https://openalex.org/keywords/continuous-function","display_name":"Continuous function (set theory)","score":0.4180302023887634},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3577239513397217},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2736102342605591}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7017274498939514},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.573176383972168},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5047277212142944},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4682784080505371},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.45410168170928955},{"id":"https://openalex.org/C2778199601","wikidata":"https://www.wikidata.org/wiki/Q5165476","display_name":"Continuous function (set theory)","level":3,"score":0.4180302023887634},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3577239513397217},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2736102342605591},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/070708937","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070708937","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W312985163","https://openalex.org/W1986143641","https://openalex.org/W2002775473","https://openalex.org/W2053990026","https://openalex.org/W2062227017","https://openalex.org/W2069912449","https://openalex.org/W2071004961","https://openalex.org/W2071952768","https://openalex.org/W2072763969","https://openalex.org/W2096613063","https://openalex.org/W2106956798","https://openalex.org/W2137224975","https://openalex.org/W2167651376","https://openalex.org/W2569481888","https://openalex.org/W3037830703","https://openalex.org/W4242477236"],"related_works":["https://openalex.org/W2385263368","https://openalex.org/W2347422947","https://openalex.org/W2061292372","https://openalex.org/W2377341410","https://openalex.org/W1454600515","https://openalex.org/W328874995","https://openalex.org/W2353392568","https://openalex.org/W2357754181","https://openalex.org/W1970439845","https://openalex.org/W1999999993"],"abstract_inverted_index":{"We":[0,35],"study":[1],"the":[2,44,53,79,155],"uniform":[3],"(Chebyshev)":[4],"approximation":[5,22],"of":[6,18,74,138],"continuous":[7],"and":[8,135],"piecewise":[9],"r-smooth":[10,183],"($r\\ge2$)":[11],"functions":[12,45,146,184],"$f:[0,T]\\to\\mathbb{R}$":[13],"with":[14,46,124,147,186],"a":[15,37,95,164,171],"finite":[16],"number":[17],"singular":[19,50,152,189],"points.":[20,34,190],"The":[21],"algorithms":[23],"use":[24],"only":[25,77,85],"n":[26,89],"function":[27,115],"values":[28],"at":[29,47],"adaptively":[30],"or":[31],"nonadaptively":[32],"chosen":[33],"construct":[36],"nonadaptive":[38,165],"algorithm":[39,121,166],"$\\mathcal{A}_{r,n}^{\\rm":[40,109,122,174],"non}$":[41,110,168],"that,":[42],"for":[43,86,131,145,181],"most":[48],"one":[49,151],"point,":[51],"enjoys":[52],"best":[54],"possible":[55],"convergence":[56,75,156],"rate":[57,73,157],"$n^{-r}$.":[58],"This":[59,140],"is":[60,98,105,170],"in":[61,78,94],"sharp":[62],"contrast":[63],"to":[64,101,107,117],"results":[65],"concerning":[66],"discontinuous":[67],"functions.":[68],"For":[69],"$r\\ge3$,":[70],"this":[71],"optimal":[72],"holds":[76],"asymptotic":[80],"sense,":[81],"i.e.,":[82],"it":[83,104],"occurs":[84],"sufficiently":[87],"large":[88],"that":[90,97,169],"depends":[91],"on":[92],"f":[93,185],"way":[96],"practically":[99],"impossible":[100],"verify.":[102],"However,":[103,154],"enough":[106],"modify":[108],"by":[111,163],"using":[112],"$(r+1)\\lfloor(r-1)/2\\rfloor$":[113],"extra-1pt":[114],"evaluations":[116],"obtain":[118],"an":[119],"adaptive":[120],"ada}$":[123],"error":[125],"satisfying":[126],"$\\|f-\\mathcal{A}_{r,n}^{\\rm":[127],"ada}f\\|_C\\le":[128],"C_rT^r\\|f^{(r)}\\|_L^\\infty":[129],"n^{-r}$":[130],"all":[132,182],"$n\\ge":[133],"n_0$":[134],"$n_0$":[136],"independent":[137],"f.":[139],"result":[141],"cannot":[142],"be":[143,160],"achieved":[144],"more":[148],"than":[149],"just":[150],"point.":[153],"$n^{-r}$":[158],"can":[159],"recovered":[161],"asymptotically":[162],"$\\overline{\\mathcal{A}}_{r,n}^{\\rm":[167],"slightly":[172],"modified":[173],"non}$.":[175],"Specifically,":[176],"$\\limsup_{n\\to\\infty}\\|f-\\overline{\\mathcal{A}}_{r,n}^{\\rm":[177],"non}f\\|_\\C\\cdot":[178],"n^r\\le":[179],"C_rT^r\\|f^{(r)}\\|_L^\\infty$":[180],"finitely":[187],"many":[188]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
