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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Tarvainen, Jenni", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-07T07:36:02Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-07T07:36:02Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95641", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutustutaan Fourier-sarjoihin ja niiden suppenemiseen. Tutkielman tarkoituksena on selvitt\u00e4\u00e4, mitk\u00e4 funktioiden ominaisuudet takaavat niiden Fourier-sarjojen suppenemisen kohti alkuper\u00e4ist\u00e4 funktiota ja mitk\u00e4 toisaalta eiv\u00e4t. Tarkastelun keski\u00f6ss\u00e4 ovat jatkuvat 2\u03c0-periodiset funktiot ja Fourier-sarjojen pisteitt\u00e4inen suppeneminen.\n\nFourier-sarjat ovat \u00e4\u00e4rett\u00f6mi\u00e4 trigonometrisi\u00e4 sarjoja, jotka m\u00e4\u00e4ritell\u00e4\u00e4n 2\u03c0-mittaisella v\u00e4lill\u00e4 m\u00e4\u00e4ritellyille Lebesgue-integroituville funktioille tai yht\u00e4pit\u00e4v\u00e4sti niiden 2\u03c0-periodisille laajennuksille. Tutkielmassa k\u00e4sitelt\u00e4v\u00e4t funktiot ovat kompleksiarvoisia, jolloin Fourier-sarjat ja -kertoimet m\u00e4\u00e4ritell\u00e4\u00e4n kompleksisten eksponenttifunktioiden avulla. \n\nAnalyysille luontaiseen tapaan Fourier-sarjojen suppenemista tutkitaan niiden osasummien muodostamien funktiojonojen avulla. T\u00e4ll\u00f6in esimerkiksi jatkuvan 2\u03c0-periodisen funktion Fourier-sarjan, joka suppenee itseisesti, osoitetaan suppenevan my\u00f6s tasaisesti kohti alkuper\u00e4ist\u00e4 funktiota. Havaittaessa, ett\u00e4 Fourier-sarjojen osasummat voidaan esitt\u00e4\u00e4 tarkasteltavan funktion ja Dirichlet-ytimen konvoluutiona, avautuu uusi n\u00e4k\u00f6kulma suppenemisen tutkimiseen. Koska jatkuvien funktioiden konvoluutiot hyvien ytimien kanssa muodostavat tasaisesti suppenevan funktiojonon, her\u00e4\u00e4 toive jatkuvien funktioiden Fourier-sarjojen suppenemisesta. Valitettavasti Dirichlet-ytimet eiv\u00e4t muodosta hyvien ytimien joukkoa, jolloin suppenemisen tutkimista t\u00e4ytyy jatkaa. Dirichlet-ytimille ja hyville ytimille osoitettujen tulosten avulla saadaan kuitenkin todistettua derivoituvien ja H\u00f6lder-jatkuvien funktioiden Fourier-sarjojen suppenevan pisteitt\u00e4in kohti tarkasteltuja funktioita.\n\nRatkaisua jatkuvien funktioiden Fourier-sarjojen suppenemiseen etsit\u00e4\u00e4n lopuksi funktionaalianalyysin perusteista. Tasaisen rajoituksen periaatteen avulla osoitetaan, ett\u00e4 on olemassa jatkuva 2\u03c0-periodinen funktio, jonka Fourier-sarja hajaantuu origossa. N\u00e4in ollen funktion jatkuvuus ei ole riitt\u00e4v\u00e4 ehto sen Fourier-sarjan pisteitt\u00e4iselle suppenemiselle kaikkialla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-06-07T07:36:02Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-07T07:36:02Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "67", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "suppeneminen", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Fourier-sarjojen suppeneminen", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202406074403", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Fourier'n sarjat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matemaattinen analyysi", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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