Approximation by antiderivatives
Identifieur interne : 000D85 ( Istex/Corpus ); précédent : 000D84; suivant : 000D86Approximation by antiderivatives
Auteurs : Wolfgang LuhSource :
- Constructive Approximation [ 0176-4276 ] ; 1986-12-01.
Abstract
Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (−n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (−n)(z)=dφ (−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: (1) For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ−nk} converges tof uniformly onB. (2) For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ−mk} converges tof compactly onU. (3) For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.
Url:
DOI: 10.1007/BF01893424
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Approximation by antiderivatives</title><author><name sortKey="Luh, Wolfgang" sort="Luh, Wolfgang" uniqKey="Luh W" first="Wolfgang" last="Luh">Wolfgang Luh</name><affiliation><mods:affiliation>Fachbereich 4/Mathematik, Universität Trier, Postfach 38 25, D-5500, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:CFF0846FE66AB974AF82D720FBB58E784D816D95</idno><date when="1986" year="1986">1986</date><idno type="doi">10.1007/BF01893424</idno><idno type="url">https://api.istex.fr/document/CFF0846FE66AB974AF82D720FBB58E784D816D95/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000D85</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000D85</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Approximation by antiderivatives</title><author><name sortKey="Luh, Wolfgang" sort="Luh, Wolfgang" uniqKey="Luh W" first="Wolfgang" last="Luh">Wolfgang Luh</name><affiliation><mods:affiliation>Fachbereich 4/Mathematik, Universität Trier, Postfach 38 25, D-5500, Trier, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Constructive Approximation</title><title level="j" type="abbrev">Constr. Approx</title><idno type="ISSN">0176-4276</idno><idno type="eISSN">1432-0940</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><date type="published" when="1986-12-01">1986-12-01</date><biblScope unit="volume">2</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="179">179</biblScope><biblScope unit="page" to="187">187</biblScope></imprint><idno type="ISSN">0176-4276</idno></series><idno type="istex">CFF0846FE66AB974AF82D720FBB58E784D816D95</idno><idno type="DOI">10.1007/BF01893424</idno><idno type="ArticleID">BF01893424</idno><idno type="ArticleID">Art12</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0176-4276</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (−n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (−n)(z)=dφ (−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: (1) For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ−nk} converges tof uniformly onB. (2) For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ−mk} converges tof compactly onU. (3) For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Wolfgang Luh</name><affiliations><json:string>Fachbereich 4/Mathematik, Universität Trier, Postfach 38 25, D-5500, Trier, Germany</json:string></affiliations></json:item></author><articleId><json:string>BF01893424</json:string><json:string>Art12</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (−n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (−n)(z)=dφ (−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: (1) For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ−nk} converges tof uniformly onB. (2) For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ−mk} converges tof compactly onU. (3) For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.</abstract><qualityIndicators><score>5.075</score><pdfVersion>1.3</pdfVersion><pdfPageSize>468 x 676.784 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>854</abstractCharCount><pdfWordCount>3335</pdfWordCount><pdfCharCount>15592</pdfCharCount><pdfPageCount>9</pdfPageCount><abstractWordCount>145</abstractWordCount></qualityIndicators><title>Approximation by antiderivatives</title><refBibs><json:item><host><author><json:item><name>G Alexits</name></json:item></author><title>Konvergenzprobleme der Orthogonalreihen</title><publicationDate>1960</publicationDate></host></json:item><json:item><author><json:item><name>G,D Birkhoff</name></json:item></author><host><volume>189</volume><pages><last>475</last><first>473</first></pages><author></author><title>C.R. Acad. Sci. 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Luh FachbereichMathematik Universit~t Trier Postfach</title><publicationDate>1975</publicationDate></host><title>Theorem on the "universality" of the Riemann zeta-function</title><publicationDate>1975</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>2</volume><pages><last>187</last><first>179</first></pages><issn><json:string>0176-4276</json:string></issn><issue>1</issue><subject><json:item><value>Analysis</value></json:item><json:item><value>Numerical Analysis</value></json:item></subject><journalId><json:string>365</json:string></journalId><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1432-0940</json:string></eissn><title>Constructive Approximation</title><publicationDate>1986</publicationDate><copyrightDate>1986</copyrightDate></host><categories><wos><json:string>science</json:string><json:string>mathematics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>mathematics & statistics</json:string><json:string>numerical & computational mathematics</json:string></scienceMetrix></categories><publicationDate>1986</publicationDate><copyrightDate>1986</copyrightDate><doi><json:string>10.1007/BF01893424</json:string></doi><id>CFF0846FE66AB974AF82D720FBB58E784D816D95</id><score>0.6255089</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/CFF0846FE66AB974AF82D720FBB58E784D816D95/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/CFF0846FE66AB974AF82D720FBB58E784D816D95/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/CFF0846FE66AB974AF82D720FBB58E784D816D95/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Approximation by antiderivatives</title><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><availability><p>Springer-Verlag New York Inc., 1986</p></availability><date>1985-07-31</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Approximation by antiderivatives</title><author xml:id="author-1"><persName><forename type="first">Wolfgang</forename><surname>Luh</surname></persName><affiliation>Fachbereich 4/Mathematik, Universität Trier, Postfach 38 25, D-5500, Trier, Germany</affiliation></author></analytic><monogr><title level="j">Constructive Approximation</title><title level="j" type="abbrev">Constr. Approx</title><idno type="journal-ID">365</idno><idno type="pISSN">0176-4276</idno><idno type="eISSN">1432-0940</idno><idno type="issue-article-count">27</idno><idno type="volume-issue-count">0</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><date type="published" when="1986-12-01"></date><biblScope unit="volume">2</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="179">179</biblScope><biblScope unit="page" to="187">187</biblScope></imprint></monogr><idno type="istex">CFF0846FE66AB974AF82D720FBB58E784D816D95</idno><idno type="DOI">10.1007/BF01893424</idno><idno type="ArticleID">BF01893424</idno><idno type="ArticleID">Art12</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1985-07-31</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (−n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (−n)(z)=dφ (−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: (1) For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ−nk} converges tof uniformly onB. (2) For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ−mk} converges tof compactly onU. (3) For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Analysis</term></item><item><term>Numerical Analysis</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1985-07-31">Created</change><change when="1986-12-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2016-11-22">References added</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-01-20">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/CFF0846FE66AB974AF82D720FBB58E784D816D95/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Springer, Publisher found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer-Verlag</PublisherName><PublisherLocation>New York</PublisherLocation></PublisherInfo><Journal><JournalInfo JournalProductType="ArchiveJournal" NumberingStyle="Unnumbered"><JournalID>365</JournalID><JournalPrintISSN>0176-4276</JournalPrintISSN><JournalElectronicISSN>1432-0940</JournalElectronicISSN><JournalTitle>Constructive Approximation</JournalTitle><JournalAbbreviatedTitle>Constr. Approx</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Analysis</JournalSubject><JournalSubject Type="Secondary">Numerical Analysis</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>2</VolumeIDStart><VolumeIDEnd>2</VolumeIDEnd><VolumeIssueCount>0</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular"><IssueInfo TocLevels="0"><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd><IssueArticleCount>27</IssueArticleCount><IssueHistory><CoverDate><DateString>1986</DateString><Year>1986</Year><Month>12</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Springer-Verlag New York Inc.</CopyrightHolderName><CopyrightYear>1986</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art12"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF01893424</ArticleID><ArticleDOI>10.1007/BF01893424</ArticleDOI><ArticleSequenceNumber>12</ArticleSequenceNumber><ArticleTitle Language="En">Approximation by antiderivatives</ArticleTitle><ArticleFirstPage>179</ArticleFirstPage><ArticleLastPage>187</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>6</Month><Day>17</Day></RegistrationDate><Received><Year>1985</Year><Month>7</Month><Day>31</Day></Received><Revised><Year>1985</Year><Month>12</Month><Day>9</Day></Revised></ArticleHistory><ArticleCopyright><CopyrightHolderName>Springer-Verlag New York Inc.</CopyrightHolderName><CopyrightYear>1986</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>365</JournalID><VolumeIDStart>2</VolumeIDStart><VolumeIDEnd>2</VolumeIDEnd><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Wolfgang</GivenName><FamilyName>Luh</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>Fachbereich 4/Mathematik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postbox>Postfach 38 25</Postbox><Postcode>D-5500</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>Let Ω ⊂<Emphasis Type="Bold">C</Emphasis> be an open set with simply connected components and suppose that the function<Emphasis Type="Italic">φ</Emphasis> is holomorphic on Ω. We prove the existence of a sequence {<Emphasis Type="Italic">φ</Emphasis><Superscript>(−n)</Superscript>} of<Emphasis Type="Italic">n</Emphasis>-fold antiderivatives (i.e., we have<Emphasis Type="Italic">φ</Emphasis><Superscript>(0)</Superscript>(<Emphasis Type="Italic">z</Emphasis>)∶=<Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">z</Emphasis>) and<Emphasis Type="Italic">φ</Emphasis><Superscript>(−n)</Superscript>(<Emphasis Type="Italic">z</Emphasis>)=<Emphasis Type="Italic">dφ</Emphasis><Superscript>(−n−1)</Superscript>(<Emphasis Type="Italic">z</Emphasis>)/<Emphasis Type="Italic">dz</Emphasis> for all<Emphasis Type="Italic">n</Emphasis> ∈ N<Subscript>0</Subscript> and z ∈ Ω) such that the following properties hold:<OrderedList><ListItem><ItemNumber>(1)</ItemNumber><ItemContent><Para>For any compact set<Emphasis Type="Italic">B</Emphasis> ⊂Ω with connected complement and any function<Emphasis Type="Italic">f</Emphasis> that is continuous on<Emphasis Type="Italic">B</Emphasis> and holomorphic in its interior, there exists a sequence {<Emphasis Type="Italic">n</Emphasis><Subscript>k</Subscript>} such that {φ<Superscript>−nk</Superscript>} converges to<Emphasis Type="Italic">f</Emphasis> uniformly on<Emphasis Type="Italic">B</Emphasis>.</Para></ItemContent></ListItem><ListItem><ItemNumber>(2)</ItemNumber><ItemContent><Para>For any open set<Emphasis Type="Italic">U</Emphasis> ⊂Ω with simply connected components and any function<Emphasis Type="Italic">f</Emphasis> that is holomorphic on<Emphasis Type="Italic">U</Emphasis>, there exists a sequence {<Emphasis Type="Italic">m</Emphasis><Subscript>k</Subscript>} such that {φ<Superscript>−mk</Superscript>} converges to<Emphasis Type="Italic">f</Emphasis> compactly on<Emphasis Type="Italic">U</Emphasis>.</Para></ItemContent></ListItem><ListItem><ItemNumber>(3)</ItemNumber><ItemContent><Para>For any measurable set<Emphasis Type="Italic">E</Emphasis> ⊂Ω and any function<Emphasis Type="Italic">f</Emphasis> that is measurable on<Emphasis Type="Italic">E</Emphasis>, there exists a sequence {<Emphasis Type="Italic">p</Emphasis><Subscript>k</Subscript>} such that {<Emphasis Type="Italic">φ</Emphasis><Superscript>(-Pk)</Superscript>} converges to<Emphasis Type="Italic">f</Emphasis> almost everywhere on<Emphasis Type="Italic">E</Emphasis>.</Para></ItemContent></ListItem></OrderedList></Para></Abstract><KeywordGroup Language="En"><Heading>AMS 1980 subject classification</Heading><Keyword>Primary, 30 E 10</Keyword><Keyword>Secondary, 30 B 60</Keyword></KeywordGroup><KeywordGroup Language="En"><Heading>Key words and phrases</Heading><Keyword>Approximation in the complex plane</Keyword><Keyword>Holomorphic functions</Keyword><Keyword>Antiderivatives</Keyword><Keyword>Uniform convergence</Keyword><Keyword>Compact convergence</Keyword><Keyword>Almost-everywhere convergence</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by Dieter Gaier.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Approximation by antiderivatives</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Approximation by antiderivatives</title></titleInfo><name type="personal"><namePart type="given">Wolfgang</namePart><namePart type="family">Luh</namePart><affiliation>Fachbereich 4/Mathematik, Universität Trier, Postfach 38 25, D-5500, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Springer-Verlag</publisher><place><placeTerm type="text">New York</placeTerm></place><dateCreated encoding="w3cdtf">1985-07-31</dateCreated><dateIssued encoding="w3cdtf">1986-12-01</dateIssued><copyrightDate encoding="w3cdtf">1986</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (−n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (−n)(z)=dφ (−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: (1) For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ−nk} converges tof uniformly onB. (2) For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ−mk} converges tof compactly onU. (3) For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.</abstract><relatedItem type="host"><titleInfo><title>Constructive Approximation</title></titleInfo><titleInfo type="abbreviated"><title>Constr. Approx</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1986-12-01</dateIssued><copyrightDate encoding="w3cdtf">1986</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Analysis</topic><topic>Numerical Analysis</topic></subject><identifier type="ISSN">0176-4276</identifier><identifier type="eISSN">1432-0940</identifier><identifier type="JournalID">365</identifier><identifier type="IssueArticleCount">27</identifier><identifier type="VolumeIssueCount">0</identifier><part><date>1986</date><detail type="volume"><number>2</number><caption>vol.</caption></detail><detail type="issue"><number>1</number><caption>no.</caption></detail><extent unit="pages"><start>179</start><end>187</end></extent></part><recordInfo><recordOrigin>Springer-Verlag New York Inc., 1986</recordOrigin></recordInfo></relatedItem><identifier type="istex">CFF0846FE66AB974AF82D720FBB58E784D816D95</identifier><identifier type="DOI">10.1007/BF01893424</identifier><identifier type="ArticleID">BF01893424</identifier><identifier type="ArticleID">Art12</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag New York Inc., 1986</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Springer-Verlag New York Inc., 1986</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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