{"id":"https://openalex.org/W3090535532","doi":"https://doi.org/10.1007/s10208-021-09504-0","title":"A New Upper Bound for Sampling Numbers","display_name":"A New Upper Bound for Sampling Numbers","publication_year":2021,"publication_date":"2021-04-26","ids":{"openalex":"https://openalex.org/W3090535532","doi":"https://doi.org/10.1007/s10208-021-09504-0","mag":"3090535532"},"language":"lv","primary_location":{"id":"doi:10.1007/s10208-021-09504-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09504-0","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09504-0.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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09504-0.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084366998","display_name":"Nicolas Nagel","orcid":"https://orcid.org/0009-0004-3362-3543"},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Nicolas Nagel","raw_affiliation_strings":["Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","Faculty of Mathematics, TU Chemnitz, Chemnitz, Germany"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]},{"raw_affiliation_string":"Faculty of Mathematics, TU Chemnitz, Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034469128","display_name":"Martin Sch\u00e4fer","orcid":"https://orcid.org/0000-0003-3670-8388"},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Martin Sch\u00e4fer","raw_affiliation_strings":["Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","Faculty of Mathematics, TU Chemnitz, Chemnitz, Germany"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]},{"raw_affiliation_string":"Faculty of Mathematics, TU Chemnitz, Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010819277","display_name":"Tino Ullrich","orcid":"https://orcid.org/0000-0002-0055-0542"},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Tino Ullrich","raw_affiliation_strings":["Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","[Chemnitz University of Technology]"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]},{"raw_affiliation_string":"[Chemnitz University of Technology]","institution_ids":["https://openalex.org/I2610724"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5084366998"],"corresponding_institution_ids":["https://openalex.org/I2610724"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.2481,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74631532,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"22","issue":"2","first_page":"445","last_page":"468"},"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":0.9987999796867371,"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":0.9952999949455261,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.745503306388855},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7175495028495789},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6745743155479431},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6039437055587769},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.5246255993843079},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5008220672607422},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.45408159494400024},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.43745312094688416},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39965009689331055},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1397622525691986}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.745503306388855},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7175495028495789},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6745743155479431},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6039437055587769},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5246255993843079},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5008220672607422},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.45408159494400024},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.43745312094688416},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39965009689331055},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1397622525691986},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10208-021-09504-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09504-0","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09504-0.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:2010.00327","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2010.00327","pdf_url":"https://arxiv.org/pdf/2010.00327","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"","raw_type":"text"},{"id":"mag:3090535532","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2010.00327.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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.2010.00327","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2010.00327","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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"}],"best_oa_location":{"id":"doi:10.1007/s10208-021-09504-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09504-0","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09504-0.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/G7697924455","display_name":null,"funder_award_id":"Ul-403/2-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8263299036","display_name":null,"funder_award_id":"DFG Ul-403/2-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320311034","display_name":"Technische Universit\u00e4t Chemnitz","ror":"https://ror.org/00a208s56"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3090535532.pdf","grobid_xml":"https://content.openalex.org/works/W3090535532.grobid-xml"},"referenced_works_count":46,"referenced_works":["https://openalex.org/W17488683","https://openalex.org/W342858109","https://openalex.org/W1493969319","https://openalex.org/W1565176583","https://openalex.org/W1642222099","https://openalex.org/W1698155719","https://openalex.org/W1820164671","https://openalex.org/W1839061065","https://openalex.org/W1867316708","https://openalex.org/W2042799176","https://openalex.org/W2047772840","https://openalex.org/W2069230279","https://openalex.org/W2079002396","https://openalex.org/W2084010106","https://openalex.org/W2095136391","https://openalex.org/W2097897435","https://openalex.org/W2107411554","https://openalex.org/W2110888545","https://openalex.org/W2148195034","https://openalex.org/W2300951034","https://openalex.org/W2324115660","https://openalex.org/W2480981092","https://openalex.org/W2556237558","https://openalex.org/W2789988695","https://openalex.org/W2798679239","https://openalex.org/W2919678159","https://openalex.org/W2963103105","https://openalex.org/W2963390618","https://openalex.org/W2964011010","https://openalex.org/W2974878803","https://openalex.org/W2991247463","https://openalex.org/W3016021445","https://openalex.org/W3017325642","https://openalex.org/W3088796800","https://openalex.org/W3092056385","https://openalex.org/W3098045837","https://openalex.org/W3110811957","https://openalex.org/W3114241195","https://openalex.org/W3114928620","https://openalex.org/W3132924785","https://openalex.org/W3155724638","https://openalex.org/W3158733253","https://openalex.org/W3178042296","https://openalex.org/W3216643960","https://openalex.org/W4214639496","https://openalex.org/W4255975175"],"related_works":["https://openalex.org/W3157488464","https://openalex.org/W3013890728","https://openalex.org/W3017325642","https://openalex.org/W2510425678","https://openalex.org/W3132857214","https://openalex.org/W2397017720","https://openalex.org/W3105711037","https://openalex.org/W2767086972","https://openalex.org/W2405151523","https://openalex.org/W2204591777","https://openalex.org/W2953320526","https://openalex.org/W3100802811","https://openalex.org/W2048176749","https://openalex.org/W2586408739","https://openalex.org/W2594412010","https://openalex.org/W2982624330","https://openalex.org/W2954888688","https://openalex.org/W2886188331","https://openalex.org/W2098385246","https://openalex.org/W2059867647"],"abstract_inverted_index":{"Abstract":[0],"We":[1,350],"provide":[2],"a":[3,37,216,222,233,239,271],"new":[4],"upper":[5],"bound":[6,214,354],"for":[7,283],"sampling":[8,226,273],"numbers":[9,168],"$$(g_n)_{n\\in":[10],"\\mathbb":[11,143],"{N}}$$":[12,144],"<mml:math":[13,55,89,145,180,295,318,339,361],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[14,56,90,146,181,296,319,340,362],"<mml:msub>":[15,18,147,150,192,200,321,368,373],"<mml:mrow>":[16,24,57,91,95,103,114,148,156,182,186,196,297,303,306,320,325,328,341,363,367,375,383],"<mml:mo>(</mml:mo>":[17,106,149,187,197,304,326],"<mml:mi>g</mml:mi>":[19,97,369],"<mml:mi>n</mml:mi>":[20,25,98,107,110,119,130,370,382],"</mml:msub>":[21,29,153,161,195,203,324,371,380],"<mml:mo>)</mml:mo>":[22,108,154,189,204,311,333],"</mml:mrow>":[23,28,63,109,121,134,138,155,160,190,205,206,308,312,313,330,334,335,347,379,386],"<mml:mo>\u2208</mml:mo>":[26,158],"<mml:mi>N</mml:mi>":[27,159],"</mml:math>":[30,64,139,162,207,314,336,348],"associated":[31],"with":[32,238,337],"the":[33,44,69,164,172,213,244,257,268,276,288,352],"compact":[34],"embedding":[35,174],"of":[36,46,166,171,225,232,247,270,291],"separable":[38],"reproducing":[39],"kernel":[40],"Hilbert":[41],"space":[42,45],"into":[43],"square":[47],"integrable":[48],"functions.":[49],"There":[50],"are":[51,66,229],"universal":[52],"constants":[53],"$$C,c&gt;0$$":[54],"<mml:mi>C</mml:mi>":[58,104,374],"<mml:mo>,</mml:mo>":[59,128,133,199,377],"<mml:mi>c</mml:mi>":[60,118],"<mml:mo>&gt;</mml:mo>":[61,343],"<mml:mn>0</mml:mn>":[62],"(which":[65],"specified":[67],"in":[68,236,315],"paper)":[70],"such":[71],"that":[72],"$$\\begin{aligned}":[73,355],"g^2_n":[74],"\\le":[75,357],"\\frac{C\\log":[76],"(n)}{n}\\sum":[77],"\\limits":[78],"_{k\\ge":[79],"\\lfloor":[80],"cn":[81],"\\rfloor":[82],"}":[83],"\\sigma":[84],"_k^2,\\quad":[85],"n\\ge":[86],"2,":[87],"\\end{aligned}$$":[88,360],"<mml:mtable>":[92,364],"<mml:mtr>":[93,365],"<mml:mtd>":[94,366],"<mml:msubsup>":[96,123,298],"<mml:mn>2</mml:mn>":[99,126,132,194,323,346],"</mml:msubsup>":[100,127,302],"<mml:mo>\u2264</mml:mo>":[101,372],"<mml:mfrac>":[102],"<mml:mo>log</mml:mo>":[105],"</mml:mfrac>":[111],"<mml:munder>":[112],"<mml:mo>\u2211</mml:mo>":[113],"<mml:mi>k</mml:mi>":[115,125,152,157],"<mml:mo>\u2265</mml:mo>":[116,131],"<mml:mo>\u230a</mml:mo>":[117],"<mml:mo>\u230b</mml:mo>":[120],"</mml:munder>":[122],"<mml:mi>\u03c3</mml:mi>":[124,151],"<mml:mspace/>":[129],"</mml:mtd>":[135],"</mml:mtr>":[136],"</mml:mtable>":[137],"where":[140],"$$(\\sigma":[141],"_k)_{k\\in":[142],"is":[163,215,262],"sequence":[165],"singular":[167],"(approximation":[169],"numbers)":[170],"Hilbert\u2013Schmidt":[173],"$$\\mathrm":[175],"{Id}:H(K)":[176],"\\rightarrow":[177],"L_2(D,\\varrho":[178],"_D)$$":[179],"<mml:mi>Id</mml:mi>":[183],"<mml:mo>:</mml:mo>":[184],"<mml:mi>H</mml:mi>":[185,299],"<mml:mi>K</mml:mi>":[188],"<mml:mo>\u2192</mml:mo>":[191],"<mml:mi>L</mml:mi>":[193,322],"<mml:mi>D</mml:mi>":[198,202],"<mml:mi>\u03f1</mml:mi>":[201],".":[208,349],"The":[209,279],"algorithm":[210,219],"which":[211,250],"realizes":[212],"least":[217],"squares":[218],"based":[220],"on":[221],"specific":[223],"set":[224],"nodes.":[227],"These":[228],"constructed":[230],"out":[231],"random":[234],"draw":[235],"combination":[237],"down-sampling":[240],"procedure":[241],"coming":[242],"from":[243],"celebrated":[245],"proof":[246],"Weaver\u2019s":[248],"conjecture,":[249],"was":[251],"shown":[252],"to":[253,256,287],"be":[254,285],"equivalent":[255],"Kadison\u2013Singer":[258],"problem.":[259],"Our":[260],"result":[261,281],"non-constructive":[263],"since":[264],"we":[265],"only":[266],"show":[267],"existence":[269],"linear":[272],"operator":[274],"realizing":[275],"above":[277],"bound.":[278],"general":[280],"can":[282],"instance":[284],"applied":[286],"well-known":[289],"situation":[290],"$$H^s_{\\text":[292],"{mix}}(\\mathbb":[293],"{T}^d)$$":[294,317],"<mml:mtext>mix</mml:mtext>":[300],"<mml:mi>s</mml:mi>":[301,342,376,385],"<mml:msup>":[305,327,381],"<mml:mi>T</mml:mi>":[307,329],"<mml:mi>d</mml:mi>":[309,331,378],"</mml:msup>":[310,332,387],"$$L_2(\\mathbb":[316],"$$s&gt;1/2$$":[338],"<mml:mn>1</mml:mn>":[344],"<mml:mo>/</mml:mo>":[345],"obtain":[351],"asymptotic":[353],"g_n":[356],"C_{s,d}n^{-s}\\log":[358],"(n)^{(d-1)s+1/2},":[359],"<mml:mo>-</mml:mo>":[384]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}