Fourier-analyysi, syksy 2012

Last modified by vpiilola@helsinki_fi on 2024/03/27 10:35

Fourier-analyysi, syksy 2012

Luennoitsija

Kari Astala

Ajankohtaista

Laajuus

10 op.

Tyyppi

Syventävä opinto

Esitietovaatimukset

Perustietoina Mitta ja integraali sekä

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org.xwiki.rendering.macro.parameter.MacroParameterException: Parameter [body] is mandatory
at org.xwiki.rendering.wikimacro.internal.DefaultWikiMacro.validate(DefaultWikiMacro.java:134)
at org.xwiki.rendering.wikimacro.internal.DefaultWikiMacro.execute(DefaultWikiMacro.java:98)
at org.xwiki.rendering.wikimacro.internal.DefaultWikiMacro.execute(DefaultWikiMacro.java:53)
at wiki:xwiki//com.xwiki.macros.internal.AbstractUnprefixedConfluenceBridgeMacro.execute(AbstractUnprefixedConfluenceBridgeMacro.java:122)
at wiki:xwiki//com.xwiki.macros.internal.MathinlineMacro.execute(MathinlineMacro.java:33)
at org.xwiki.rendering.internal.transformation.macro.MacroTransformation.transform(MacroTransformation.java:311)
at org.xwiki.rendering.internal.transformation.DefaultRenderingContext.transformInContext(DefaultRenderingContext.java:183)
at org.xwiki.rendering.internal.transformation.DefaultTransformationManager.performTransformations(DefaultTransformationManager.java:88)
at org.xwiki.display.internal.DocumentContentAsyncExecutor.executeInCurrentExecutionContext(DocumentContentAsyncExecutor.java:395)
at org.xwiki.display.internal.DocumentContentAsyncExecutor.execute(DocumentContentAsyncExecutor.java:268)
at org.xwiki.display.internal.DocumentContentAsyncRenderer.execute(DocumentContentAsyncRenderer.java:112)
at org.xwiki.rendering.async.internal.block.AbstractBlockAsyncRenderer.render(AbstractBlockAsyncRenderer.java:157)
at org.xwiki.rendering.async.internal.block.AbstractBlockAsyncRenderer.render(AbstractBlockAsyncRenderer.java:54)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.syncRender(DefaultAsyncRendererExecutor.java:290)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.render(DefaultAsyncRendererExecutor.java:267)
at org.xwiki.rendering.async.internal.block.DefaultBlockAsyncRendererExecutor.execute(DefaultBlockAsyncRendererExecutor.java:125)
at org.xwiki.display.internal.DocumentContentDisplayer.display(DocumentContentDisplayer.java:67)
at org.xwiki.display.internal.DocumentContentDisplayer.display(DocumentContentDisplayer.java:43)
at org.xwiki.display.internal.DefaultDocumentDisplayer.display(DefaultDocumentDisplayer.java:96)
at org.xwiki.display.internal.DefaultDocumentDisplayer.display(DefaultDocumentDisplayer.java:39)
at org.xwiki.sheet.internal.SheetDocumentDisplayer.display(SheetDocumentDisplayer.java:123)
at org.xwiki.sheet.internal.SheetDocumentDisplayer.display(SheetDocumentDisplayer.java:52)
at org.xwiki.display.internal.ConfiguredDocumentDisplayer.display(ConfiguredDocumentDisplayer.java:68)
at org.xwiki.display.internal.ConfiguredDocumentDisplayer.display(ConfiguredDocumentDisplayer.java:42)
at com.xpn.xwiki.doc.XWikiDocument.display(XWikiDocument.java:1368)
at com.xpn.xwiki.doc.XWikiDocument.getRenderedContent(XWikiDocument.java:1505)
at com.xpn.xwiki.doc.XWikiDocument.displayDocument(XWikiDocument.java:1454)
at com.xpn.xwiki.doc.XWikiDocument.displayDocument(XWikiDocument.java:1423)
at com.xpn.xwiki.api.Document.displayDocument(Document.java:787)
at jdk.internal.reflect.GeneratedMethodAccessor585.invoke(Unknown Source)
at java.base/jdk.internal.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.base/java.lang.reflect.Method.invoke(Method.java:569)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.doInvoke(UberspectImpl.java:571)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.invoke(UberspectImpl.java:554)
at org.apache.velocity.runtime.parser.node.ASTMethod.execute(ASTMethod.java:221)
at org.apache.velocity.runtime.parser.node.ASTReference.execute(ASTReference.java:368)
at org.apache.velocity.runtime.parser.node.ASTReference.value(ASTReference.java:704)
at org.apache.velocity.runtime.parser.node.ASTExpression.value(ASTExpression.java:75)
at org.apache.velocity.runtime.parser.node.ASTSetDirective.render(ASTSetDirective.java:242)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.apache.velocity.runtime.parser.node.SimpleNode.render(SimpleNode.java:439)
at org.apache.velocity.runtime.parser.node.ASTIfStatement.render(ASTIfStatement.java:190)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.xwiki.velocity.internal.directive.TryCatchDirective.render(TryCatchDirective.java:86)
at org.apache.velocity.runtime.parser.node.ASTDirective.render(ASTDirective.java:304)
at org.apache.velocity.runtime.parser.node.SimpleNode.render(SimpleNode.java:439)
at org.apache.velocity.Template.merge(Template.java:358)
at org.apache.velocity.Template.merge(Template.java:262)
at org.xwiki.velocity.internal.InternalVelocityEngine.evaluate(InternalVelocityEngine.java:225)
at com.xpn.xwiki.internal.template.VelocityTemplateEvaluator.evaluateContent(VelocityTemplateEvaluator.java:105)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.evaluateContent(TemplateAsyncRenderer.java:219)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.renderVelocity(TemplateAsyncRenderer.java:174)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:135)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:54)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.lambda$syncRender$0(DefaultAsyncRendererExecutor.java:284)
at com.xpn.xwiki.internal.security.authorization.DefaultAuthorExecutor.call(DefaultAuthorExecutor.java:98)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.syncRender(DefaultAsyncRendererExecutor.java:284)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.render(DefaultAsyncRendererExecutor.java:267)
at org.xwiki.rendering.async.internal.block.DefaultBlockAsyncRendererExecutor.render(DefaultBlockAsyncRendererExecutor.java:154)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:904)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderFromSkin(InternalTemplateManager.java:866)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:853)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderNoException(InternalTemplateManager.java:808)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderNoException(InternalTemplateManager.java:800)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.renderNoException(DefaultTemplateManager.java:79)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.renderNoException(DefaultTemplateManager.java:73)
at org.xwiki.template.script.TemplateScriptService.render(TemplateScriptService.java:54)
at jdk.internal.reflect.GeneratedMethodAccessor238.invoke(Unknown Source)
at java.base/jdk.internal.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.base/java.lang.reflect.Method.invoke(Method.java:569)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.doInvoke(UberspectImpl.java:571)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.invoke(UberspectImpl.java:554)
at org.apache.velocity.runtime.parser.node.ASTMethod.execute(ASTMethod.java:221)
at org.apache.velocity.runtime.parser.node.ASTReference.execute(ASTReference.java:368)
at org.apache.velocity.runtime.parser.node.ASTReference.render(ASTReference.java:492)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.apache.velocity.runtime.directive.VelocimacroProxy.render(VelocimacroProxy.java:218)
at org.apache.velocity.runtime.directive.RuntimeMacro.render(RuntimeMacro.java:331)
at org.apache.velocity.runtime.directive.RuntimeMacro.render(RuntimeMacro.java:261)
at org.apache.velocity.runtime.parser.node.ASTDirective.render(ASTDirective.java:304)
at org.apache.velocity.runtime.parser.node.SimpleNode.render(SimpleNode.java:439)
at org.apache.velocity.Template.merge(Template.java:358)
at org.apache.velocity.Template.merge(Template.java:262)
at org.xwiki.velocity.internal.InternalVelocityEngine.evaluate(InternalVelocityEngine.java:225)
at com.xpn.xwiki.internal.template.VelocityTemplateEvaluator.evaluateContent(VelocityTemplateEvaluator.java:105)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.evaluateContent(TemplateAsyncRenderer.java:219)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.renderVelocity(TemplateAsyncRenderer.java:174)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:135)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:54)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.lambda$syncRender$0(DefaultAsyncRendererExecutor.java:284)
at com.xpn.xwiki.internal.security.authorization.DefaultAuthorExecutor.call(DefaultAuthorExecutor.java:98)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.syncRender(DefaultAsyncRendererExecutor.java:284)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.render(DefaultAsyncRendererExecutor.java:267)
at org.xwiki.rendering.async.internal.block.DefaultBlockAsyncRendererExecutor.render(DefaultBlockAsyncRendererExecutor.java:154)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:904)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderFromSkin(InternalTemplateManager.java:866)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:853)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderNoException(InternalTemplateManager.java:808)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderNoException(InternalTemplateManager.java:800)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.renderNoException(DefaultTemplateManager.java:79)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.renderNoException(DefaultTemplateManager.java:73)
at org.xwiki.template.script.TemplateScriptService.render(TemplateScriptService.java:54)
at jdk.internal.reflect.GeneratedMethodAccessor238.invoke(Unknown Source)
at java.base/jdk.internal.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.base/java.lang.reflect.Method.invoke(Method.java:569)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.doInvoke(UberspectImpl.java:571)
at org.apache.velocity.util.introspection.UberspectImpl$VelMethodImpl.invoke(UberspectImpl.java:554)
at org.apache.velocity.runtime.parser.node.ASTMethod.execute(ASTMethod.java:221)
at org.apache.velocity.runtime.parser.node.ASTReference.execute(ASTReference.java:368)
at org.apache.velocity.runtime.parser.node.ASTReference.render(ASTReference.java:492)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.apache.velocity.runtime.directive.VelocimacroProxy.render(VelocimacroProxy.java:218)
at org.apache.velocity.runtime.directive.RuntimeMacro.render(RuntimeMacro.java:331)
at org.apache.velocity.runtime.directive.RuntimeMacro.render(RuntimeMacro.java:261)
at org.apache.velocity.runtime.parser.node.ASTDirective.render(ASTDirective.java:304)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.apache.velocity.runtime.parser.node.ASTIfStatement.render(ASTIfStatement.java:171)
at org.apache.velocity.runtime.parser.node.ASTBlock.render(ASTBlock.java:147)
at org.apache.velocity.runtime.parser.node.SimpleNode.render(SimpleNode.java:439)
at org.apache.velocity.runtime.parser.node.ASTIfStatement.render(ASTIfStatement.java:190)
at org.apache.velocity.runtime.parser.node.SimpleNode.render(SimpleNode.java:439)
at org.apache.velocity.Template.merge(Template.java:358)
at org.apache.velocity.Template.merge(Template.java:262)
at org.xwiki.velocity.internal.InternalVelocityEngine.evaluate(InternalVelocityEngine.java:225)
at com.xpn.xwiki.internal.template.VelocityTemplateEvaluator.evaluateContent(VelocityTemplateEvaluator.java:105)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.evaluateContent(TemplateAsyncRenderer.java:219)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.renderVelocity(TemplateAsyncRenderer.java:174)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:135)
at com.xpn.xwiki.internal.template.TemplateAsyncRenderer.render(TemplateAsyncRenderer.java:54)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.lambda$syncRender$0(DefaultAsyncRendererExecutor.java:284)
at com.xpn.xwiki.internal.security.authorization.DefaultAuthorExecutor.call(DefaultAuthorExecutor.java:98)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.syncRender(DefaultAsyncRendererExecutor.java:284)
at org.xwiki.rendering.async.internal.DefaultAsyncRendererExecutor.render(DefaultAsyncRendererExecutor.java:267)
at org.xwiki.rendering.async.internal.block.DefaultBlockAsyncRendererExecutor.render(DefaultBlockAsyncRendererExecutor.java:154)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:904)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderFromSkin(InternalTemplateManager.java:866)
at com.xpn.xwiki.internal.template.InternalTemplateManager.renderFromSkin(InternalTemplateManager.java:846)
at com.xpn.xwiki.internal.template.InternalTemplateManager.render(InternalTemplateManager.java:832)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.render(DefaultTemplateManager.java:91)
at com.xpn.xwiki.internal.template.DefaultTemplateManager.render(DefaultTemplateManager.java:85)
at com.xpn.xwiki.XWiki.evaluateTemplate(XWiki.java:2565)
at com.xpn.xwiki.web.Utils.parseTemplate(Utils.java:180)
at com.xpn.xwiki.web.XWikiAction.execute(XWikiAction.java:651)
at com.xpn.xwiki.web.XWikiAction.execute(XWikiAction.java:339)
at com.xpn.xwiki.web.LegacyActionServlet.service(LegacyActionServlet.java:108)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:623)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:210)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at com.xpn.xwiki.web.ActionFilter.doFilter(ActionFilter.java:122)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.wysiwyg.filter.ConversionFilter.doFilter(ConversionFilter.java:61)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:51)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.resource.servlet.RoutingFilter.doFilter(RoutingFilter.java:132)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.container.servlet.filters.internal.SavedRequestRestorerFilter.doFilter(SavedRequestRestorerFilter.java:208)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.container.servlet.filters.internal.SetCharacterEncodingFilter.doFilter(SetCharacterEncodingFilter.java:111)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.apache.catalina.core.ApplicationDispatcher.invoke(ApplicationDispatcher.java:661)
at org.apache.catalina.core.ApplicationDispatcher.processRequest(ApplicationDispatcher.java:425)
at org.apache.catalina.core.ApplicationDispatcher.doForward(ApplicationDispatcher.java:357)
at org.apache.catalina.core.ApplicationDispatcher.forward(ApplicationDispatcher.java:294)
at com.xpn.xwiki.web.XWikiAction.redirectSpaceURLs(XWikiAction.java:1172)
at com.xpn.xwiki.web.XWikiAction.execute(XWikiAction.java:509)
at com.xpn.xwiki.web.XWikiAction.execute(XWikiAction.java:339)
at com.xpn.xwiki.web.LegacyActionServlet.service(LegacyActionServlet.java:108)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:623)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:210)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at com.xpn.xwiki.web.ActionFilter.doFilter(ActionFilter.java:122)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.wysiwyg.filter.ConversionFilter.doFilter(ConversionFilter.java:61)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:51)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.container.servlet.filters.internal.SetHTTPHeaderFilter.doFilter(SetHTTPHeaderFilter.java:63)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.resource.servlet.RoutingFilter.doFilter(RoutingFilter.java:132)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.container.servlet.filters.internal.SavedRequestRestorerFilter.doFilter(SavedRequestRestorerFilter.java:208)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.xwiki.container.servlet.filters.internal.SetCharacterEncodingFilter.doFilter(SetCharacterEncodingFilter.java:111)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:179)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:154)
at org.apache.catalina.core.StandardWrapperValve.invoke(StandardWrapperValve.java:168)
at org.apache.catalina.core.StandardContextValve.invoke(StandardContextValve.java:90)
at org.apache.catalina.authenticator.AuthenticatorBase.invoke(AuthenticatorBase.java:481)
at org.apache.catalina.core.StandardHostValve.invoke(StandardHostValve.java:130)
at org.apache.catalina.valves.ErrorReportValve.invoke(ErrorReportValve.java:93)
at org.apache.catalina.valves.StuckThreadDetectionValve.invoke(StuckThreadDetectionValve.java:185)
at org.apache.catalina.valves.AbstractAccessLogValve.invoke(AbstractAccessLogValve.java:670)
at org.apache.catalina.core.StandardEngineValve.invoke(StandardEngineValve.java:74)
at org.apache.catalina.connector.CoyoteAdapter.service(CoyoteAdapter.java:346)
at org.apache.coyote.ajp.AjpProcessor.service(AjpProcessor.java:424)
at org.apache.coyote.AbstractProcessorLight.process(AbstractProcessorLight.java:63)
at org.apache.coyote.AbstractProtocol$ConnectionHandler.process(AbstractProtocol.java:928)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.doRun(NioEndpoint.java:1786)
at org.apache.tomcat.util.net.SocketProcessorBase.run(SocketProcessorBase.java:52)
at org.apache.tomcat.util.threads.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1191)
at org.apache.tomcat.util.threads.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:659)
at org.apache.tomcat.util.threads.TaskThread$WrappingRunnable.run(TaskThread.java:63)
at java.base/java.lang.Thread.run(Thread.java:840)

-avaruuksien ominaisuudet kurssilta Reaalianalyysi I.

Lisätietoja I. Holopaisen luentomonisteessa: Reaalianalyysi I

Lisäksi tarvitaan joitakin tietoja funktionaalianalyysistä, joita voidaan tarvittaessa käydä lyhyesti läpi.

Luentoajat

Viikot 36-42 ja 44-50 ti 10-12, ke 12-14 C123.

Jälkimmäisen periodin ensimmäinen luento tiistaina 30. lokakuuta.

Kurssiin sisältyy laskuharjoituksia 2 viikkotuntia, joita pitää Antti Perälä, tiistaisin klo 16-18 salissa B322. Ensimmäiset harjoitukset Ti 11.9.

Kokeet

Kaksi kurssikoetta, Pe 19.10. ja Ma 17.12. Molemmat kokeet ovat klo 13 -15.

Ensimmäisen kurssikokeen koealue: Luvut I - VII, eli sivut 1-65 luentomuistiinpanoista ja Harjoitukset 1 - 6.

Ohessa ensimmäisen Kurssikokeen tulokset (koe 19.10.2012) ja Kurssikokeen ratkaisut.

Koepisteiden keskiarvo oli 18,3.

(max = 24).

Toisen kurssikokeen alue: Luvut VIII - XIV.1, eli sivut 66 -129 luentomuistiinpanoista, ja Harjoitukset 7-12.

Ohessa toisen kurssikokeen tulokset.

Ohessa toisen kurssikokeen ratkaisut.

Kurssin sisällöstä

Kurssi on Fourier analyysin peruskurssi; sitä suositellaan sekä matematiikan että sovelletun matematiikan opiskelijoille.

Kurssin ensimmäinen periodi käsittelee lähinnä Fourier sarjoja, esim. niiden

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-teoriaa ja pisteittäistä konvergenssia sekä Fourier-sarjojen sovelluksia mm. aalto- ja lämpöyhtälöiden alkuarvo-ongelmiin, lukuteoriaan tai geometriaan.

Fourier sarjojen jälkeen siirrymme jatkuvaan Fourier-muunnokseen, aluksi integroituvien (eli

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-) funktioiden tapauksessa, ja sen jälkeen

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- sekä muissa

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-avaruuksissa. Tämä taas johdattaa luontevasti distribuutioihin ja niiden Fourier-muunnoksiin.

Fourier-muunnosta voidaan puolestaan käyttää apuna kun tutkimme erilaisia integraali-ja
differentiaalioperaattoreita kuten Hilbert-muunnosta, tai löytämään vakiokertoimisten
differentiaalioperaattoreiden perusratkaisuja. Fourier-muunnos on tärkeä apuväline monella matematiikan ja sen sovellusten alueella, siihen perustuu esim. keskeinen raja-arvolause stokastiikassa. Kurssilla perehdymme myös diskreettiin Fourier muunnokseen ja nopean Fourier muunnoksen (FFT) perusominaisuuksiin.

Luentomuistiinpanot

Muistiinpanot: Luennot.FourierAnalyysi2012.pdf

Kirjallisuus

Seuraavia kirjoja voi soveltuvin osin käyttää kurssin tukena.

Stein-Shakarchi: Fourier Analysis, An Introduction.
Rudin: Real and Complex Analysis.

Lisätietoja Fourier analyysistä löytää esim. seuraavista kirjoista.

Grafakos: Classical and Modern Fourier Analysis.
Rudin: Functional Analysis.
Duoandikoetxea: Fourier Analysis.

Ilmoittaudu

Unohditko ilmoittautua? Mitä tehdä.

Laskuharjoitustehtävät

Harjoitus 1
Harjoitus 2
Harjoitus 3
Harjoitus 4
Harjoitus 5
Harjoitus 6
Harjoitus 7
Harjoitus 8
Harjoitus 9
Harjoitus 10
Harjoitus 11
Harjoitus 12

Laskuharjoitukset

Harjoitusten perusteella saa lisäpisteitä seuraavasti:

25% = +1p, 35% = +2p, 45% = +3p, 55% = +4p, 65% = +5p ja 75% = +6p.

Ryhmä	Päivä	Aika	Paikka	Pitäjä
1.	ti	16-18	B322	Antti Perälä

Fourier-analyysi, syksy 2012

Fourier-analyysi, syksy 2012

Luennoitsija

Ajankohtaista

Tyyppi

Esitietovaatimukset

Luentoajat

Kokeet

Kurssin sisällöstä

Luentomuistiinpanot

Kirjallisuus

Ilmoittaudu

Laskuharjoitustehtävät

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