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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Tuominen, Heli", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kullaa, Hilla", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-12-07T06:47:31Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-12-07T06:47:31Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/60515", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 ty\u00f6ss\u00e4 tutustutaan kahteen Weierstrassin tulokseen. Karl Wilhelm Theodor Weierstrass oli saksalainen matemaatikko (1815-1897). Monelle Weierstrassin nimi on tuttu Bolzano-Weierstrassin lauseesta tai Weierstrassin M-testist\u00e4. H\u00e4n my\u00f6s muotoili (\u03b5, \u03b4)m\u00e4\u00e4ritelm\u00e4n jatkuvuudelle. T\u00e4ss\u00e4 tutkielmassa keskityt\u00e4\u00e4n kuitenkin kahteen approksimaatioteorian tulokseen. N\u00e4iden kahden Weierstrassin tuloksen voidaan ajatella olevan approksimaatioteorian klassisia perustuloksia.\nEnsimm\u00e4inen tulos on vuodelta 1872. Se on Weierstrassin esimerkki jatkuvasta ei miss\u00e4\u00e4n pisteess\u00e4 derivoituvasta funktiosta. K\u00e4ytt\u00e4m\u00e4ll\u00e4 funktiosarjoja Weierstrass konstruoi funktion, joka on jatkuva, mutta ei miss\u00e4\u00e4n pisteess\u00e4 derivoituva. T\u00e4t\u00e4 kutsutaan Weierstrassin funktioksi. Jatkuvien, ei miss\u00e4\u00e4n pisteess\u00e4 derivoituvien funktioiden l\u00f6ytyminen mahdollisti monen sovelluksen syntymisen kuten Brownin liike, fraktaalit ja kaaosteoria.\nToinen tulos on vuodelta 1885. Kyseess\u00e4 on Weierstrassin approksimaatiolause. Lauseen mukaan jokaista jatkuvaa funktiota reaalilukujen joukossa voidaan approksimoida mielivaltaisen tarkasti sup-normissa polynomeja k\u00e4ytt\u00e4en.\nTutkielmassa l\u00e4hdet\u00e4\u00e4n liikkeelle m\u00e4\u00e4rittelem\u00e4ll\u00e4 aputuloksia ja k\u00e4ym\u00e4ll\u00e4 l\u00e4pi ty\u00f6ss\u00e4 k\u00e4ytett\u00e4vi\u00e4 merkint\u00f6j\u00e4. Ty\u00f6 etenee funktiosarjojen ja potenssisarjojen k\u00e4sittelyll\u00e4. T\u00e4ll\u00f6in esitell\u00e4\u00e4n ja todistetaan my\u00f6s Raaben testi, joka on sarjan suppenemistesti. Sen avulla pystyt\u00e4\u00e4n tutkimaan suppeneeko potenssisarja suppenemisv\u00e4lins\u00e4 p\u00e4\u00e4tepisteiss\u00e4. Raaben testi\u00e4 tarvitaan Weiertrassin approksimaatiolauseen todistamisessa. Ty\u00f6ss\u00e4 todistus tehd\u00e4\u00e4n kahdella eri tavalla. Ensimm\u00e4inen todistus tehd\u00e4\u00e4n Lebesguen tavalla ja toinen niin sanotun konvoluutioapproksimaation avulla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2018-12-07T06:47:31Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-12-07T06:47:31Z (GMT). No. of bitstreams: 0\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "32", "language": "", "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": "approksimaatiolause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ei miss\u00e4\u00e4n derivoituva funktio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Raaben testi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Weierstrass", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Weierstrassin lause ja muita approksimaatiotuloksia", "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-201812075022", "language": "", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "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": "", "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": "funktiot", "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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