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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Annala, Leevi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-06-20T07:46:51Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-06-20T07:46:51Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1703951", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/54594", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Funktion f Fourier'n sarja on \u00e4\u00e4ret\u00f6n funktiosarja, jossa summataan funktiosta f ja summausindeksist\u00e4 n riippuvia Fourier'n kertoimia funktiolla e^{inx} kerrottuna. Fourier'n sarjoja k\u00e4ytet\u00e4\u00e4n esimerkiksi osittaisdifferentiaaliyht\u00e4l\u00f6iden ratkaisemiseen. \n\nT\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n Fourier'n sarjan suppenemista. Kun Fourier'n sarja keksittiin, pitk\u00e4\u00e4n luultiin, ett\u00e4 jatkuvan funktion Fourier'n sarja suppenee aina. T\u00e4ss\u00e4 ty\u00f6ss\u00e4 osoitetaan, ett\u00e4 n\u00e4in ei ole.\n\nEnsin ty\u00f6ss\u00e4 osoitetaan, ett\u00e4 jatkuvan funktion Fourier'n sarja \"melkein suppenee,\" eli on Abel- ja Ces\u00e0ro-summautuva. Abel-summautuvuudessa sarjan summattavat kerrotaan luvulla r^n, miss\u00e4 luku r on itseisarvoltaan pienempi kuin 1 ja n kertoo monesko summattava on kyseess\u00e4, ja tutkitaan suppeneeko n\u00e4in saatu sarja \u00e4\u00e4ret\u00f6n sarja, jonka summattavana on r^n a_n. Ces\u00e0ro-summautuvuudessa puolestaan lasketaan osasummien keskiarvoja, ja tutkitaan suppeneeko osasummien keskiarvojen jono.\n\nLis\u00e4ksi todistetaan, ett\u00e4 kun funktio on rajoitetusti heilahteleva, niin sen Fourier'n sarja suppenee niiss\u00e4 pisteiss\u00e4, miss\u00e4 funktio on jatkuva. T\u00e4m\u00e4 tarkoittaa samalla sit\u00e4, ett\u00e4 kun funktio on paloittain C^1-funktio, Lipschitz-jatkuva tai absoluuttisesti jatkuva, niin funktion Fourier'n sarja suppenee. \n\nViimeisen\u00e4 ty\u00f6ss\u00e4 esitell\u00e4\u00e4n jatkuva funktio, jonka Fourier'n sarja hajaantuu. Funktion konstruoinnissa k\u00e4ytet\u00e4\u00e4n menetelm\u00e4\u00e4, jossa kansanomaisesti sanottuna pienet ongelmat kasaantuvat ja tuottavat massiivisia ongelmia.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Leevi Annala (lealanna) on 2017-06-20 07:46:51.042096. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-06-20T07:46:51Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201706202965.pdf: 417017 bytes, checksum: 449032560e571f88abd096c2d845e0c0 (MD5)\nlicense.html: 4783 bytes, checksum: b1696d4f708a97532491c6d2500378b2 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-06-20T07:46:51Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201706202965.pdf: 417017 bytes, checksum: 449032560e571f88abd096c2d845e0c0 (MD5)\nlicense.html: 4783 bytes, checksum: b1696d4f708a97532491c6d2500378b2 (MD5)\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (77 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "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": "matematiikka", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "analyysi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Fourier", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "suppeneminen", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Fourier'n sarjan suppeneminen", "language": null, "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-201706202965", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2017-06-20T07:46:51Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "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": "fi", "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": "analyysi", "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.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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