{"id":"https://openalex.org/W2034008227","doi":"https://doi.org/10.1090/s0025-5718-2011-02549-4","title":"On the convergence rates of Legendre approximation","display_name":"On the convergence rates of Legendre approximation","publication_year":2011,"publication_date":"2011-10-18","ids":{"openalex":"https://openalex.org/W2034008227","doi":"https://doi.org/10.1090/s0025-5718-2011-02549-4","mag":"2034008227"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2011-02549-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02549-4","pdf_url":"https://www.ams.org/mcom/2012-81-278/S0025-5718-2011-02549-4/S0025-5718-2011-02549-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2012-81-278/S0025-5718-2011-02549-4/S0025-5718-2011-02549-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070318438","display_name":"Haiyong Wang","orcid":"https://orcid.org/0000-0002-0360-4489"},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]},{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE","CN"],"is_corresponding":false,"raw_author_name":"Haiyong Wang","raw_affiliation_strings":["Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u00e2\u0080\u0099s Republic of China","Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium","Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I139660479"]},{"raw_affiliation_string":"Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium","institution_ids":["https://openalex.org/I99464096"]},{"raw_affiliation_string":"Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u2019s Republic of China","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002179101","display_name":"Shuhuang Xiang","orcid":"https://orcid.org/0000-0002-6727-6170"},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shuhuang Xiang","raw_affiliation_strings":["Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u00e2\u0080\u0099s Republic of China","Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I139660479"]},{"raw_affiliation_string":"Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People\u2019s Republic of China","institution_ids":["https://openalex.org/I139660479"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":9.1319,"has_fulltext":true,"cited_by_count":169,"citation_normalized_percentile":{"value":0.98701299,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"81","issue":"278","first_page":"861","last_page":"877"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9987999796867371,"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"}},"topics":[{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9987999796867371,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9909999966621399,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9887999892234802,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.9147339463233948},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8327619433403015},{"id":"https://openalex.org/keywords/legendre-function","display_name":"Legendre function","score":0.7196128368377686},{"id":"https://openalex.org/keywords/associated-legendre-polynomials","display_name":"Associated Legendre polynomials","score":0.6557421684265137},{"id":"https://openalex.org/keywords/barycentric-coordinate-system","display_name":"Barycentric coordinate system","score":0.647907018661499},{"id":"https://openalex.org/keywords/legendres-equation","display_name":"Legendre's equation","score":0.6398890018463135},{"id":"https://openalex.org/keywords/legendre-wavelet","display_name":"Legendre wavelet","score":0.6352606415748596},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.559270977973938},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5352122783660889},{"id":"https://openalex.org/keywords/lagrange-polynomial","display_name":"Lagrange polynomial","score":0.5001425743103027},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.485532283782959},{"id":"https://openalex.org/keywords/spherical-harmonics","display_name":"Spherical harmonics","score":0.4662092626094818},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4351130723953247},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.4137412905693054},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.3163166642189026},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.22949963808059692},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.16020378470420837},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12436974048614502},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.0997469425201416}],"concepts":[{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.9147339463233948},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8327619433403015},{"id":"https://openalex.org/C153262748","wikidata":"https://www.wikidata.org/wiki/Q17098449","display_name":"Legendre function","level":3,"score":0.7196128368377686},{"id":"https://openalex.org/C36956377","wikidata":"https://www.wikidata.org/wiki/Q228376","display_name":"Associated Legendre polynomials","level":5,"score":0.6557421684265137},{"id":"https://openalex.org/C128356825","wikidata":"https://www.wikidata.org/wiki/Q738422","display_name":"Barycentric coordinate system","level":2,"score":0.647907018661499},{"id":"https://openalex.org/C86287605","wikidata":"https://www.wikidata.org/wiki/Q4454970","display_name":"Legendre's equation","level":3,"score":0.6398890018463135},{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.6352606415748596},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.559270977973938},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5352122783660889},{"id":"https://openalex.org/C64208722","wikidata":"https://www.wikidata.org/wiki/Q861606","display_name":"Lagrange polynomial","level":3,"score":0.5001425743103027},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.485532283782959},{"id":"https://openalex.org/C3768446","wikidata":"https://www.wikidata.org/wiki/Q877100","display_name":"Spherical harmonics","level":2,"score":0.4662092626094818},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4351130723953247},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.4137412905693054},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.3163166642189026},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.22949963808059692},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.16020378470420837},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12436974048614502},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.0997469425201416},{"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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C502989409","wikidata":"https://www.wikidata.org/wiki/Q11425","display_name":"Animation","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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.0},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.0},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-2011-02549-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02549-4","pdf_url":"https://www.ams.org/mcom/2012-81-278/S0025-5718-2011-02549-4/S0025-5718-2011-02549-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2011-02549-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02549-4","pdf_url":"https://www.ams.org/mcom/2012-81-278/S0025-5718-2011-02549-4/S0025-5718-2011-02549-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1025573998","display_name":"\u7535\u3001\u78c1\u3001\u58f0\u5b66\u7814\u7a76\u4e2d\u4e00\u7c7b\u9ad8\u632f\u8361\u79ef\u5206\u65b9\u7a0b\u3001\u5fae\u5206\u65b9\u7a0b\u6e10\u8fdb\u7406\u8bba\u53ca\u5176\u9ad8\u6027\u80fd\u8ba1\u7b97\u7814\u7a76","funder_award_id":"11071260","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320326217","display_name":"Hunan Provincial Innovation Foundation for Postgraduate","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2034008227.pdf","grobid_xml":"https://content.openalex.org/works/W2034008227.grobid-xml"},"referenced_works_count":51,"referenced_works":["https://openalex.org/W106404678","https://openalex.org/W120757762","https://openalex.org/W140628048","https://openalex.org/W149575133","https://openalex.org/W222318538","https://openalex.org/W280536263","https://openalex.org/W315472779","https://openalex.org/W434478127","https://openalex.org/W1490180844","https://openalex.org/W1558122728","https://openalex.org/W1603910097","https://openalex.org/W1635622277","https://openalex.org/W1805381866","https://openalex.org/W1912930600","https://openalex.org/W1971099220","https://openalex.org/W1971587363","https://openalex.org/W1976782593","https://openalex.org/W1977463378","https://openalex.org/W1981115560","https://openalex.org/W2017836554","https://openalex.org/W2020328455","https://openalex.org/W2031567600","https://openalex.org/W2033022456","https://openalex.org/W2044788680","https://openalex.org/W2052169332","https://openalex.org/W2056522847","https://openalex.org/W2059998776","https://openalex.org/W2061622006","https://openalex.org/W2073732659","https://openalex.org/W2082108265","https://openalex.org/W2082849992","https://openalex.org/W2095817904","https://openalex.org/W2142567026","https://openalex.org/W2144181355","https://openalex.org/W2159105505","https://openalex.org/W2313299119","https://openalex.org/W2314359380","https://openalex.org/W2316418224","https://openalex.org/W2410898212","https://openalex.org/W2479395228","https://openalex.org/W2622633536","https://openalex.org/W3031888853","https://openalex.org/W3038165843","https://openalex.org/W3150423993","https://openalex.org/W4254246953","https://openalex.org/W4285719527","https://openalex.org/W4295022726","https://openalex.org/W4299412321","https://openalex.org/W4300846967","https://openalex.org/W4301973099","https://openalex.org/W4402490215"],"related_works":["https://openalex.org/W2953361964","https://openalex.org/W3005861762","https://openalex.org/W3200178185","https://openalex.org/W1988418937","https://openalex.org/W3130427480","https://openalex.org/W2022538265","https://openalex.org/W2288349649","https://openalex.org/W2053826141","https://openalex.org/W4296546724","https://openalex.org/W2034418652"],"abstract_inverted_index":{"The":[0],"problem":[1],"of":[2,5,7,18,31,67,81],"the":[3,15,19,22,32,37,54,59,82],"rate":[4],"convergence":[6],"Legendre":[8,23,34,44,83,89],"approximation":[9,45],"is":[10],"considered.":[11],"We":[12],"first":[13],"establish":[14],"decay":[16],"rates":[17],"coefficients":[20],"in":[21,36,53,65],"series":[24,35],"expansion":[25],"and":[26,70],"then":[27],"derive":[28],"error":[29],"bounds":[30],"truncated":[33],"uniform":[38],"norm.":[39],"In":[40,48],"addition,":[41],"we":[42,50],"consider":[43],"with":[46],"interpolation.":[47],"particular,":[49],"are":[51,74,92],"interested":[52],"barycentric":[55,63],"Lagrange":[56],"formula":[57],"at":[58],"Gauss-Legendre":[60,68],"points.":[61],"Explicit":[62],"weights,":[64,73],"terms":[66],"points":[69],"corresponding":[71],"quadrature":[72],"presented":[75],"that":[76],"allow":[77],"a":[78],"fast":[79],"evaluation":[80],"interpolation":[84,90],"formula.":[85],"Error":[86],"estimates":[87],"for":[88],"polynomials":[91],"also":[93],"given.":[94]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":13},{"year":2022,"cited_by_count":14},{"year":2021,"cited_by_count":14},{"year":2020,"cited_by_count":19},{"year":2019,"cited_by_count":15},{"year":2018,"cited_by_count":12},{"year":2017,"cited_by_count":11},{"year":2016,"cited_by_count":11},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":9},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":6}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}