{"id":"https://openalex.org/W7220696002","doi":"https://doi.org/10.48550/arxiv.2610.06645","title":"Quantitative $\u0393$-Convergence: A Variational Extrapolation Framework","display_name":"Quantitative $\u0393$-Convergence: A Variational Extrapolation Framework","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220696002","doi":"https://doi.org/10.48550/arxiv.2610.06645"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06645","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06645","pdf_url":"https://arxiv.org/pdf/2610.06645","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.06645","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020994805","display_name":"\u00d3scar Dom\u00ednguez","orcid":"https://orcid.org/0000-0001-7751-1632","observed_orcids":["https://orcid.org/0000-0001-7751-1632"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dominguez, Oscar","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5081865508","display_name":"Mario Milman","orcid":"https://orcid.org/0000-0003-3735-8009","observed_orcids":["https://orcid.org/0000-0003-3735-8009"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Milman, Mario","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.45969998836517334,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.45969998836517334,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.350600004196167,"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.05310000106692314,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rudin-osher-fatemi-model","display_name":"Rudin-Osher-Fatemi model","score":0.9819999933242798},{"id":"https://openalex.org/keywords/image-processing","display_name":"image processing","score":0.6060000061988831},{"id":"https://openalex.org/keywords/bourgain-brezis-mironescu-formula","display_name":"Bourgain-Brezis-Mironescu formula","score":0.9259999990463257},{"id":"https://openalex.org/keywords/heat-kernel","display_name":"heat kernel","score":0.46000000834465027}],"concepts":[{"id":"https://openalex.org/C132459708","wikidata":"https://www.wikidata.org/wiki/Q744069","display_name":"Extrapolation","level":2,"score":0.90829998254776},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7645999789237976},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.6869999766349792},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5608999729156494},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5374000072479248},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5030999779701233},{"id":"https://openalex.org/C27156116","wikidata":"https://www.wikidata.org/wiki/Q1778098","display_name":"Pointwise convergence","level":3,"score":0.4352000057697296},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3873000144958496},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.3864000141620636},{"id":"https://openalex.org/C97985569","wikidata":"https://www.wikidata.org/wiki/Q216861","display_name":"Calculus of variations","level":2,"score":0.3797999918460846},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.3691999912261963},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.36660000681877136},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.34869998693466187},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3292999863624573},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3271999955177307},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.3109000027179718},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3061000108718872},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.28940001130104065},{"id":"https://openalex.org/C523394659","wikidata":"https://www.wikidata.org/wiki/Q17086892","display_name":"Variational analysis","level":2,"score":0.2849999964237213},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.28119999170303345},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.25279998779296875},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06645","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06645","pdf_url":"https://arxiv.org/pdf/2610.06645","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,"listed_in":[],"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":"doi:10.48550/arxiv.2610.06645","doi":"https://doi.org/10.48550/arxiv.2610.06645","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06645","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.06645","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06645","pdf_url":"https://arxiv.org/pdf/2610.06645","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,"listed_in":[],"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":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G745419941","display_name":null,"funder_award_id":"PID2023","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G7572614111","display_name":null,"funder_award_id":"152878NB-I00","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"}],"funders":[{"id":"https://openalex.org/F4320322930","display_name":"Ministerio de Ciencia e Innovaci\u00f3n","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220696002.pdf","grobid_xml":"https://content.openalex.org/works/W7220696002.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W3093493291","https://openalex.org/W7125415050"],"related_works":["https://openalex.org/W2013311047","https://openalex.org/W2052178414","https://openalex.org/W2085859127","https://openalex.org/W1517403807","https://openalex.org/W4411642809","https://openalex.org/W653336373","https://openalex.org/W2135030169","https://openalex.org/W1966304071","https://openalex.org/W2767725113","https://openalex.org/W1972231406"],"abstract_inverted_index":{"This":[0],"paper":[1],"introduces":[2],"a":[3,14,55,73,114,141,199,235],"new":[4,39],"extrapolation":[5,43,79,229],"framework":[6],"within":[7],"the":[8,19,42,87,91,101,117,126,132,136,152,165,188,239],"Calculus":[9,240],"of":[10,18,22,44,53,68,125,135,145,151,154,169,187,201,230,241],"Variations.":[11,242],"We":[12],"develop":[13],"robust":[15],"quantitative":[16,236],"theory":[17],"$\u0393$-convergence":[20],"approach":[21,237],"De":[23,218],"Giorgi":[24],"by":[25,163],"combining":[26],"it":[27],"with":[28,222],"Gagliardo's":[29,223],"work":[30,233],"on":[31,211],"relative":[32],"completions":[33],"in":[34,97,110,149,198],"interpolation":[35],"theory.":[36],"A":[37],"central":[38],"tool":[40],"is":[41,113,147,160],"\\emph{variational":[45],"$K$-functionals},":[46],"which":[47],"associates":[48],"to":[49,93,116,122,238],"an":[50],"arbitrary":[51],"family":[52],"functionals":[54],"dynamic":[56],"penalty":[57],"method":[58],"that":[59,131,139,203],"simultaneously":[60],"characterizes":[61],"their":[62],"$\u0393$-limits":[63],"and":[64,104,173,179,185,190,228],"yields":[65],"explicit":[66],"rates":[67],"convergence":[69,146,184,221],"for":[70],"minimizers.":[71],"As":[72],"technical":[74],"foundation":[75],"we":[76,89,129],"prove":[77],"compactness":[78],"theorems":[80],"also":[81],"involving":[82],"Gagliardo":[83],"completions.":[84],"To":[85],"illustrate":[86],"mechanism,":[88],"apply":[90],"technique":[92],"variational":[94,155,220,226],"models":[95],"arising":[96],"image":[98,138],"processing,":[99],"notably":[100],"Rudin--Osher--Fatemi":[102],"(ROF)":[103],"Chan--Esedoglu":[105],"models.":[106],"Our":[107],"main":[108],"result":[109],"this":[111,232],"setting":[112],"contribution":[115],"\\emph{inverse":[118],"regularity":[119,134],"problem}":[120],"associated":[121],"discrete":[123],"approximations":[124],"ROF":[127],"model:":[128],"show":[130],"required":[133],"observed":[137],"guarantees":[140],"prescribed":[142],"optimal":[143],"rate":[144],"measured":[148],"terms":[150],"decay":[153],"$K$-functionals.":[156],"Surprisingly,":[157],"our":[158,176],"characterization":[159],"explicitly":[161],"captured":[162],"using":[164],"recently":[166],"introduced":[167],"spaces":[168,210],"Brezis,":[170],"Van":[171],"Schaftingen,":[172],"Yung.":[174],"Moreover,":[175],"methods":[177],"unify,":[178],"considerably":[180],"extend,":[181],"classical":[182],"pointwise":[183],"$\u0393$-versions":[186],"Bourgain--Brezis--Mironescu":[189],"Maz'ya--Shaposhnikova":[191],"formulae,":[192],"as":[193,195],"well":[194],"related":[196],"results":[197],"variety":[200],"settings":[202],"involve":[204],"BMO-type":[205],"functionals,":[206],"heat":[207],"kernels,":[208],"Sobolev":[209],"metric":[212],"measure":[213],"spaces,":[214],"...":[215],"By":[216],"bridging":[217],"Giorgi's":[219],"completion":[224],"through":[225],"$K$-functionals":[227],"compactness,":[231],"establishes":[234]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}