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These studies are available for the following degree students:</p><ul><li>Bachelor's Degree Programme in Mathematics</li><li>Master's Degree Programme in Mathematics</li><li>Bachelor's Degree Programme in Mathematics (Subject Teacher)</li><li>Master's Degree Programme in Mathematics&nbsp;(Subject Teacher)</li><li>Bachelor's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Master's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Doctoral Programme in Mathematics and Statistics</li><li>Doctoral Programme in Mathematics and Science (Specialication in Mathematics)</li><br></ul>","fi":"<p>Tämä opintojakso on tarjolla Matematiikan syventävät opinnot -ristiinopiskeluverkostossa. Verkoston opinnot ovat tarjolla seuraaville opiskelijoille:</p><ul><li>Matematiikan kandidaattiohjelma</li><li>Matematiikan maisteriohjelma</li><li>Matematiikan aineenopettajien kandidaattiohjelma</li><li>Matematiikan aineenopettajien maisteriohjelma</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan kandidaattiohjelma (matematiikan opintosuunta)</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan maisteriiohjelma (matematiikan opintosuunta)</li><li>Matematiikan ja tilastotieteen tohtoriohjelma</li><li>Matemaattisten tieteiden ja luonnontieteiden tohtoriohjelma (matematiikan opintosuunta)</li><br></ul>"},"cooperationNetwork":{"abbreviation":"matematiikansyventavat","name":{"en":"Cross-institutional studies in advanced courses in mathematics and statistics","fi":"Matematiikan ja tilastotieteen syventävien kurssien ristiinopiskelu","sv":"Korsstudier i fördjupade kurser i matematik och statistik"}}}],"gradeScaleId":"sis-0-5","outcomes":{"en":"Upon completion of the course, the student will be able to:\r\n•\tDefine Lebesgue spaces and describe their basic properties.\r\n•\tCompute the Fourier transform of functions in the Schwartz space.\r\n•\tPerform basic operations on tempered distributions, including differentiation and Fourier transform.\r\n•\tDefine and calculate weak derivatives, and construct Sobolev spaces of integer and fractional order.\r\n•\tApply Sobolev embedding theorems to analyze smoothness and regularity of functions.","fi":"Opintojakson suoritettuaan opiskelija osaa:\r\n•\tmääritellä Lebesguen avaruudet ja tuntee niiden perusominaisuudet\r\n•\tlaskea Schwartzin funktioiden Fourier muunnoksia\r\n•\ttemperoitujen distribuutioiden perusominaisuudet, kuten derivaatan ja Fourier muunnoksen laskemisen\r\n•\tmääritellä ja laskea heikkoja derivaattoja, sekä muodostaa kokonais- ja murtoasteisia Sobolevin avaruuksia\r\n•\tkäyttää Sobolevin upotuslauseita funktioiden sileyden ja säännöllisyyden tutkimiseen."},"tweetText":null,"content":{"en":"Lebesgue spaces, Schwartz spaces, Fourier transform, tempered distributions, Sobolev spaces","fi":"Lebesguen avaruudet, Schwartzin avaruudet, Fourier muunnos, temperoidut distribuutiot, Sobolevin avaruudet"},"additional":{"en":"Course is aimed to students in mathematical sciences (applied and computational mathematics).","fi":"Kurssi soveltuu matemaattisten alojen opiskelijoille (sovellettu ja laskennallinen matematiikka)."},"prerequisites":{"en":"Complex analysis 1, Calculus 2, Measure theory","fi":"Kompleksianalyysi 1, Moniulotteinen analyysi, Mittateoria"},"compulsoryFormalPrerequisites":[],"recommendedFormalPrerequisites":[],"literature":[],"learningMaterial":null,"completionMethods":[]}],"prerequisiteCourseUnit":[],"prerequisiteModule":[]},"prerequisiteCourseUnitPage":{"nodes":[]},"prerequisiteModulePage":{"nodes":[]},"parentModulePage":{"nodes":[]}},"pageContext":{"type":"courseUnit","locale":"en","title":"Introduction to function spaces","id":"otm-7849770c-2b65-39a9-b5d1-156528fac0e2","code":"DP00BD62","prerequisiteCourseUnitIds":[],"prerequisiteModuleIds":[],"parentModuleIds":[],"curriculumPeriodStartDate":"2026-08-01","curriculumPeriodEndDate":"2027-08-01","coordinatingOrgIds":[],"searchable":false,"searchTags":null,"organisationIds":["otm-71fd8cc0-9fb4-38e1-bbb0-d1d63f911ddf"],"organisations":[],"attainmentLanguages":["en","fi"],"hasSummerStudies":false,"teachingPeriods":[],"cooperationNetworkDirection":"INBOUND","hasCooperationNetworkSettings":true,"hasAvoinTeaching":false}}}