{"id":"https://openalex.org/W2090926880","doi":"https://doi.org/10.1007/s10444-009-9142-7","title":"The missing Wendland functions","display_name":"The missing Wendland functions","publication_year":2009,"publication_date":"2009-09-23","ids":{"openalex":"https://openalex.org/W2090926880","doi":"https://doi.org/10.1007/s10444-009-9142-7","mag":"2090926880"},"language":"en","primary_location":{"id":"doi:10.1007/s10444-009-9142-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9142-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9142-7.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9142-7.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085097073","display_name":"Robert Schaback","orcid":"https://orcid.org/0000-0001-7828-4355"},"institutions":[{"id":"https://openalex.org/I74656192","display_name":"University of G\u00f6ttingen","ror":"https://ror.org/01y9bpm73","country_code":"DE","type":"education","lineage":["https://openalex.org/I74656192"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Robert Schaback","raw_affiliation_strings":["Institut f\u00fcr Numerische und Angewandte Mathematik, Universit\u00e4t G\u00f6ttingen, Lotzestrasse 16\u201318, 37083, G\u00f6ttingen, Germany","Institut f\u00fcr Numerische und Angewandte Mathematik, Universit\u00e4t G\u00f6ttingen, G\u00f6ttingen, Germany 37083#TAB#"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Numerische und Angewandte Mathematik, Universit\u00e4t G\u00f6ttingen, Lotzestrasse 16\u201318, 37083, G\u00f6ttingen, Germany","institution_ids":["https://openalex.org/I74656192"]},{"raw_affiliation_string":"Institut f\u00fcr Numerische und Angewandte Mathematik, Universit\u00e4t G\u00f6ttingen, G\u00f6ttingen, Germany 37083#TAB#","institution_ids":["https://openalex.org/I74656192"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5085097073"],"corresponding_institution_ids":["https://openalex.org/I74656192"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":7.9517,"has_fulltext":true,"cited_by_count":61,"citation_normalized_percentile":{"value":0.97398271,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"34","issue":"1","first_page":"67","last_page":"81"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9889000058174133,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9728000164031982,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8798017501831055},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.7233337163925171},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6217465996742249},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5343325138092041},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.5180090069770813},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5107316970825195},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.49706652760505676},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.49360740184783936},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4306448698043823},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3741919994354248},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26939016580581665}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8798017501831055},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.7233337163925171},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6217465996742249},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5343325138092041},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.5180090069770813},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5107316970825195},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.49706652760505676},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.49360740184783936},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4306448698043823},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3741919994354248},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26939016580581665},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10444-009-9142-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9142-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9142-7.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.150.6797","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.6797","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.num.math.uni-goettingen.de/schaback/research/papers/TMWFrev1.pdf","raw_type":"text"},{"id":"pmh:oai:goescholar:1/5986","is_oa":true,"landing_page_url":"http://resolver.sub.uni-goettingen.de/purl?gs-1/5986","pdf_url":null,"source":{"id":"https://openalex.org/S4306401634","display_name":"GoeScholar  The Publication Server of the Georg-August-Universit\u00e4t G\u00f6ttingen (Georg-August-Universit\u00e4t G\u00f6ttingen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210122495","host_organization_name":"Asklepios Klinik St. Georg","host_organization_lineage":["https://openalex.org/I4210122495"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"journalArticle"},{"id":"pmh:oai:publications.goettingen-research-online.de:2/24201","is_oa":true,"landing_page_url":"https://resolver.sub.uni-goettingen.de/purl?gro-2/24201","pdf_url":null,"source":{"id":"https://openalex.org/S4306401634","display_name":"GoeScholar  The Publication Server of the Georg-August-Universit\u00e4t G\u00f6ttingen (Georg-August-Universit\u00e4t G\u00f6ttingen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210122495","host_organization_name":"Asklepios Klinik St. Georg","host_organization_lineage":["https://openalex.org/I4210122495"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"journal_article"}],"best_oa_location":{"id":"doi:10.1007/s10444-009-9142-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9142-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9142-7.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2090926880.pdf","grobid_xml":"https://content.openalex.org/works/W2090926880.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W436320942","https://openalex.org/W624213318","https://openalex.org/W1486164486","https://openalex.org/W1605182429","https://openalex.org/W1968414728","https://openalex.org/W1971597992","https://openalex.org/W1977042664","https://openalex.org/W2000158450","https://openalex.org/W2003078952","https://openalex.org/W2022863449","https://openalex.org/W2082100828","https://openalex.org/W2089245375","https://openalex.org/W2106388318","https://openalex.org/W2127689830","https://openalex.org/W2330755614"],"related_works":["https://openalex.org/W3081363737","https://openalex.org/W2049628683","https://openalex.org/W1985787639","https://openalex.org/W2383844134","https://openalex.org/W3093305317","https://openalex.org/W2374198922","https://openalex.org/W2889599571","https://openalex.org/W1833822726","https://openalex.org/W2798137860","https://openalex.org/W1998603635"],"abstract_inverted_index":{"The":[0],"Wendland":[1,53],"radial":[2],"basis":[3],"functions":[4,54,88],"(Wendland,":[5],"Adv":[6],"Comput":[7],"Math":[8],"4:389\u2013396,":[9],"1995)":[10],"are":[11,22],"piecewise":[12],"polynomial":[13],"compactly":[14],"supported":[15],"reproducing":[16,62],"kernels":[17],"in":[18,46,66,108,121],"Hilbert":[19],"spaces":[20,33,45,65],"which":[21],"norm\u2013equivalent":[23],"to":[24,73],"Sobolev":[25,32,64],"spaces.":[26],"But":[27],"they":[28,70],"only":[29],"cover":[30],"the":[31,42,51,97,103,117],"1":[34],"$$\\label{eqstartrep}":[35],"H^{d/2+k+1/2}({\\mathbf{R}}^d),\\;k\\in":[36],"{\\mathbf{N}}":[37],"$$":[38],"and":[39,59,81,126],"leave":[40],"out":[41,72],"integer":[43],"order":[44],"even":[47,60,67],"dimensions.":[48],"We":[49],"derive":[50],"missing":[52],"working":[55],"for":[56],"half-integer":[57],"k":[58],"dimensions,":[61,68],"integer-order":[63],"but":[69],"turn":[71],"have":[74],"two":[75,111],"additional":[76],"non-polynomial":[77],"terms:":[78],"a":[79,82,89,93],"logarithm":[80],"square":[83],"root.":[84],"To":[85],"give":[86],"these":[87],"solid":[90],"mathematical":[91],"foundation,":[92],"generalized":[94],"version":[95],"of":[96,110,123,130],"\u201cdimension":[98],"walk\u201d":[99],"is":[100],"applied.":[101],"While":[102],"classical":[104],"dimension":[105],"walk":[106],"proceeds":[107,120],"steps":[109,122],"space":[112],"dimensions":[113,125],"taking":[114],"single":[115,124],"derivatives,":[116],"new":[118],"one":[119],"uses":[127],"\u201chalved\u201d":[128],"derivatives":[129],"fractional":[131],"calculus.":[132]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-14T08:04:32.555800","created_date":"2025-10-10T00:00:00"}