{"id":"https://openalex.org/W2014551121","doi":"https://doi.org/10.1137/060666639","title":"Walsh Spaces Containing Smooth Functions and Quasi\u2013Monte Carlo Rules of Arbitrary High Order","display_name":"Walsh Spaces Containing Smooth Functions and Quasi\u2013Monte Carlo Rules of Arbitrary High Order","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2014551121","doi":"https://doi.org/10.1137/060666639","mag":"2014551121"},"language":"en","primary_location":{"id":"doi:10.1137/060666639","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060666639","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":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1304.0328","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004856556","display_name":"Josef Dick","orcid":"https://orcid.org/0000-0003-0142-6022"},"institutions":[{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Josef Dick","raw_affiliation_strings":["School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia","institution_ids":["https://openalex.org/I31746571"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5004856556"],"corresponding_institution_ids":["https://openalex.org/I31746571"],"apc_list":null,"apc_paid":null,"fwci":11.7929,"has_fulltext":true,"cited_by_count":138,"citation_normalized_percentile":{"value":0.98913043,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"46","issue":"3","first_page":"1519","last_page":"1553"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":1.0,"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":1.0,"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/T12923","display_name":"Digital Image Processing Techniques","score":0.953499972820282,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9143999814987183,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/mathematics","display_name":"Mathematics","score":0.8058450222015381},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.748583197593689},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.7034143805503845},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.6341159343719482},{"id":"https://openalex.org/keywords/quasi-monte-carlo-method","display_name":"Quasi-Monte Carlo method","score":0.5950952768325806},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5766755938529968},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.5313544869422913},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5192070603370667},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.49715402722358704},{"id":"https://openalex.org/keywords/walsh-function","display_name":"Walsh function","score":0.47228819131851196},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4437550902366638},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.4108359217643738},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.342048317193985},{"id":"https://openalex.org/keywords/hybrid-monte-carlo","display_name":"Hybrid Monte Carlo","score":0.2339884638786316},{"id":"https://openalex.org/keywords/markov-chain-monte-carlo","display_name":"Markov chain Monte Carlo","score":0.1181793212890625},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1060040295124054}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8058450222015381},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.748583197593689},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.7034143805503845},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.6341159343719482},{"id":"https://openalex.org/C63320529","wikidata":"https://www.wikidata.org/wiki/Q7269435","display_name":"Quasi-Monte Carlo method","level":5,"score":0.5950952768325806},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5766755938529968},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.5313544869422913},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5192070603370667},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.49715402722358704},{"id":"https://openalex.org/C164319782","wikidata":"https://www.wikidata.org/wiki/Q1934233","display_name":"Walsh function","level":2,"score":0.47228819131851196},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4437550902366638},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.4108359217643738},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.342048317193985},{"id":"https://openalex.org/C13153151","wikidata":"https://www.wikidata.org/wiki/Q1639846","display_name":"Hybrid Monte Carlo","level":4,"score":0.2339884638786316},{"id":"https://openalex.org/C111350023","wikidata":"https://www.wikidata.org/wiki/Q1191869","display_name":"Markov chain Monte Carlo","level":3,"score":0.1181793212890625},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1060040295124054},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/060666639","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060666639","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"},{"id":"pmh:oai:arXiv.org:1304.0328","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1304.0328","pdf_url":"https://arxiv.org/pdf/1304.0328","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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1304.0328","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1304.0328","pdf_url":"https://arxiv.org/pdf/1304.0328","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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W87334110","https://openalex.org/W225736130","https://openalex.org/W1557786636","https://openalex.org/W1579063588","https://openalex.org/W1607458528","https://openalex.org/W1971639230","https://openalex.org/W1971947347","https://openalex.org/W1976420124","https://openalex.org/W1976544585","https://openalex.org/W1980840110","https://openalex.org/W1981743146","https://openalex.org/W1986280275","https://openalex.org/W1992068214","https://openalex.org/W1999024885","https://openalex.org/W2004534011","https://openalex.org/W2007578934","https://openalex.org/W2008009962","https://openalex.org/W2009231757","https://openalex.org/W2010651950","https://openalex.org/W2022681869","https://openalex.org/W2024626809","https://openalex.org/W2027669629","https://openalex.org/W2030404880","https://openalex.org/W2035463901","https://openalex.org/W2037134469","https://openalex.org/W2037767367","https://openalex.org/W2043419861","https://openalex.org/W2048312356","https://openalex.org/W2057989705","https://openalex.org/W2062842007","https://openalex.org/W2065960076","https://openalex.org/W2066442888","https://openalex.org/W2073732659","https://openalex.org/W2079366547","https://openalex.org/W2086236484","https://openalex.org/W2091862355","https://openalex.org/W2093625674","https://openalex.org/W2099375630","https://openalex.org/W2126004257","https://openalex.org/W2140262981","https://openalex.org/W2145169372","https://openalex.org/W2318856246","https://openalex.org/W2333786694","https://openalex.org/W2481449583","https://openalex.org/W2499620708","https://openalex.org/W3102328113","https://openalex.org/W3135037104","https://openalex.org/W4300580367","https://openalex.org/W4301866048"],"related_works":["https://openalex.org/W2082240576","https://openalex.org/W3016438207","https://openalex.org/W2127997631","https://openalex.org/W2488809685","https://openalex.org/W1492314125","https://openalex.org/W4253135066","https://openalex.org/W4253505826","https://openalex.org/W2000121903","https://openalex.org/W1969136379","https://openalex.org/W2048544541"],"abstract_inverted_index":{"We":[0],"define":[1],"a":[2,27],"Walsh":[3,42,46],"space":[4,47],"which":[5,98],"contains":[6],"all":[7],"functions":[8],"whose":[9],"partial":[10],"mixed":[11],"derivatives":[12],"up":[13],"to":[14],"order":[15],"$\\delta":[16],"\\ge":[17],"1$":[18],"exist":[19],"and":[20],"have":[21],"finite":[22],"variation.":[23],"In":[24],"particular,":[25],"for":[26,69,79,109],"suitable":[28],"choice":[29],"of":[30,63,65,74,82,87,103,105],"parameters,":[31],"this":[32,45],"implies":[33],"that":[34,51],"certain":[35],"Sobolev":[36],"spaces":[37],"are":[38,90],"contained":[39],"in":[40],"these":[41],"spaces.":[43],"For":[44],"we":[48],"then":[49],"show":[50],"quasi\u2013Monte":[52,95],"Carlo":[53,96],"rules":[54,97],"based":[55],"on":[56],"digital":[57,88],"$(t,\\alpha,s)$-sequences":[58,89],"achieve":[59,99],"the":[60,66,80,100,106],"optimal":[61,78,101],"rate":[62,73,102],"convergence":[64,75,104],"worst-case":[67],"error":[68,108],"numerical":[70],"integration.":[71],"This":[72],"is":[76],"also":[77],"subspace":[81],"smooth":[83,111],"functions.":[84,112],"Explicit":[85],"constructions":[86],"given,":[91],"hence":[92],"providing":[93],"explicit":[94],"integration":[107],"arbitrarily":[110]},"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":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":19},{"year":2015,"cited_by_count":12},{"year":2014,"cited_by_count":17},{"year":2013,"cited_by_count":10},{"year":2012,"cited_by_count":8}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}