{"id":"https://openalex.org/W2974878803","doi":"https://doi.org/10.1007/s10208-020-09481-w","title":"Function Values Are Enough for $$L_2$$-Approximation","display_name":"Function Values Are Enough for $$L_2$$-Approximation","publication_year":2020,"publication_date":"2020-12-07","ids":{"openalex":"https://openalex.org/W2974878803","doi":"https://doi.org/10.1007/s10208-020-09481-w","mag":"2974878803"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-020-09481-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09481-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09481-w.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09481-w.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070459947","display_name":"David Krieg","orcid":"https://orcid.org/0000-0001-8180-8906"},"institutions":[{"id":"https://openalex.org/I121883995","display_name":"Johannes Kepler University of Linz","ror":"https://ror.org/052r2xn60","country_code":"AT","type":"education","lineage":["https://openalex.org/I121883995"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"David Krieg","raw_affiliation_strings":["Institut f\u00fcr Analysis, Johannes Kepler Universit\u00e4t, Linz, Austria","JOHANNES KEPLER UNIVERSITY LINZ"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Analysis, Johannes Kepler Universit\u00e4t, Linz, Austria","institution_ids":["https://openalex.org/I121883995"]},{"raw_affiliation_string":"JOHANNES KEPLER UNIVERSITY LINZ","institution_ids":["https://openalex.org/I121883995"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088907596","display_name":"Mario Ullrich","orcid":"https://orcid.org/0000-0003-1120-8467"},"institutions":[{"id":"https://openalex.org/I121883995","display_name":"Johannes Kepler University of Linz","ror":"https://ror.org/052r2xn60","country_code":"AT","type":"education","lineage":["https://openalex.org/I121883995"]},{"id":"https://openalex.org/I19880235","display_name":"Lomonosov Moscow State University","ror":"https://ror.org/010pmpe69","country_code":"RU","type":"education","lineage":["https://openalex.org/I19880235"]}],"countries":["AT","RU"],"is_corresponding":true,"raw_author_name":"Mario Ullrich","raw_affiliation_strings":["Institut f\u00fcr Analysis, Johannes Kepler Universit\u00e4t, Linz, Austria","Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University,  Moscow, Russia","JOHANNES KEPLER UNIVERSITY LINZ"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Analysis, Johannes Kepler Universit\u00e4t, Linz, Austria","institution_ids":["https://openalex.org/I121883995"]},{"raw_affiliation_string":"Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University,  Moscow, Russia","institution_ids":["https://openalex.org/I19880235"]},{"raw_affiliation_string":"JOHANNES KEPLER UNIVERSITY LINZ","institution_ids":["https://openalex.org/I121883995"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5088907596"],"corresponding_institution_ids":["https://openalex.org/I121883995","https://openalex.org/I19880235"],"apc_list":{"value":2290,"currency":"USD","value_usd":2290},"apc_paid":{"value":2290,"currency":"USD","value_usd":2290},"fwci":3.0404,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.91184074,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"21","issue":"4","first_page":"1141","last_page":"1151"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9998999834060669,"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":0.9998999834060669,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9932000041007996,"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/T10522","display_name":"Medical Imaging Techniques and Applications","score":0.9905999898910522,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7215772867202759},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.631349503993988},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6296994686126709},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5904494524002075},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.5439237952232361},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.537594735622406},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5272563695907593},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4928600788116455},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4791354835033417},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.42077624797821045},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3651053309440613},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.24656984210014343}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7215772867202759},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.631349503993988},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6296994686126709},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5904494524002075},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.5439237952232361},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.537594735622406},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5272563695907593},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4928600788116455},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4791354835033417},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.42077624797821045},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3651053309440613},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.24656984210014343},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10208-020-09481-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09481-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09481-w.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1905.02516","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1905.02516","pdf_url":"https://arxiv.org/pdf/1905.02516","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2974878803","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1905.02516.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1905.02516","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1905.02516","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s10208-020-09481-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09481-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09481-w.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8199110932","display_name":null,"funder_award_id":"F5513-N26","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2974878803.pdf","grobid_xml":"https://content.openalex.org/works/W2974878803.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W5088572","https://openalex.org/W108609485","https://openalex.org/W1493969319","https://openalex.org/W1504437704","https://openalex.org/W1820164671","https://openalex.org/W2047772840","https://openalex.org/W2069230279","https://openalex.org/W2095136391","https://openalex.org/W2300951034","https://openalex.org/W2321775062","https://openalex.org/W2497677836","https://openalex.org/W2914279594","https://openalex.org/W2919678159","https://openalex.org/W2963103105","https://openalex.org/W2963709371","https://openalex.org/W2964343121","https://openalex.org/W2991247463","https://openalex.org/W3017325642","https://openalex.org/W3178042296","https://openalex.org/W4214639496","https://openalex.org/W4302564868"],"related_works":["https://openalex.org/W3178042296","https://openalex.org/W3157200198","https://openalex.org/W3152829224","https://openalex.org/W2950054287","https://openalex.org/W1713410551","https://openalex.org/W2959342769","https://openalex.org/W1993274408","https://openalex.org/W2991247463","https://openalex.org/W2063089937","https://openalex.org/W3012485420","https://openalex.org/W2522132274","https://openalex.org/W2491158392","https://openalex.org/W2767507454","https://openalex.org/W2047562547","https://openalex.org/W1642453141","https://openalex.org/W2008739175","https://openalex.org/W2368203662","https://openalex.org/W2164084989","https://openalex.org/W2950501260","https://openalex.org/W130123586"],"abstract_inverted_index":{"Abstract":[0],"We":[1,70],"study":[2],"the":[3,16,20,30,43,50,116,121,126,143],"$$L_2$$":[4],"L":[5],"2":[6,96,174],"-approximation":[7],"of":[8,61],"functions":[9],"from":[10],"a":[11,47,94],"Hilbert":[12],"space":[13],"and":[14,129,175,184],"compare":[15],"sampling":[17,24,117],"numbers":[18,118,128,145],"with":[19,38,58,120,165],"approximation":[21,44,127,144],"numbers.":[22],"The":[23],"number":[25,45],"$$e_n$$":[26],"e":[27,83,195],"n":[28,39,48,59,84,88,93,104,106,110,196,198],"is":[29,49],"minimal":[31,51],"worst-case":[32,52],"error":[33,53],"that":[34,54,72,115,131],"can":[35,55],"be":[36,56],"achieved":[37,57],"function":[40,132],"values,":[41],"whereas":[42],"$$a_n$$":[46],"pieces":[60],"arbitrary":[62,139],"linear":[63,140],"information":[64,141],"(like":[65],"derivatives":[66],"or":[67],"Fourier":[68],"coefficients).":[69],"show":[71],"$$\\begin{aligned}":[73,187],"e_n":[74,188],"\\,\\lesssim":[75,189],"\\,":[76,190],"\\sqrt{\\frac{1}{k_n}":[77],"\\sum":[78],"_{j\\ge":[79],"k_n}":[80],"a_j^2},":[81],"\\end{aligned}$$":[82,194],"\u2272":[85,197],"1":[86,172],"k":[87,92,103],"\u2211":[89],"j":[90,95],"\u2265":[91],",":[97,183],"where":[98],"$$k_n":[99],"\\asymp":[100],"n/\\log":[101],"(n)$$":[102],"\u224d":[105],"/":[107,173],"log":[108],"(":[109,161],")":[111,164],".":[112],"This":[113],"proves":[114],"decay":[119],"same":[122],"polynomial":[123],"rate":[124],"as":[125,136,138],"therefore":[130],"values":[133],"are":[134,146],"basically":[135],"powerful":[137],"if":[142],"square-summable.":[147],"Our":[148],"result":[149],"applies,":[150],"in":[151],"particular,":[152],"to":[153],"Sobolev":[154],"spaces":[155],"$$H^s_\\mathrm{mix}(\\mathbb":[156],"{T}^d)$$":[157],"H":[158],"mix":[159],"s":[160,170],"T":[162],"d":[163,180],"dominating":[166],"mixed":[167],"smoothness":[168],"$$s>1/2$$":[169],">":[171],"dimension":[176],"$$d\\in":[177],"\\mathbb":[178],"{N}$$":[179],"\u2208":[181],"N":[182],"we":[185],"obtain":[186],"n^{-s}":[191],"\\log":[192],"^{sd}(n).":[193]},"counts_by_year":[{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":7}],"updated_date":"2026-08-23T07:36:19.812096","created_date":"2025-10-10T00:00:00"}
