utm an introduction to wavelets through linear algebra

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1、PrefaceMathematicsmajorsatMichiganStateUniversitytakea“Capstone”courseneartheendoftheirundergraduatecareers.Thecontentofthiscoursevarieswitheachoffering.Itspurposeistobringtogetherdifferenttopicsfromtheundergraduatecurriculumandintroducestudentstoade

2、velopingareainmathematics.ThistextwasoriginallywrittenforaCapstonecourse.Basicwavelettheoryisanaturaltopicforsuchacourse.Byname,waveletsdatebackonlytothe1980s.Ontheboundarybetweenmathematicsandengineering,wavelettheoryshowsstudentsthatmathematicsrese

3、archisstillthriving,withimportantapplicationsinareassuchasimagecompressionandthenumericalsolutionofdifferentialequations.Theauthorbelievesthattheessentialsofwavelettheoryaresufficientlyelementarytobetaughtsuccessfullytoadvancedundergraduates.Thistexti

4、sintendedforundergraduates,soonlyabasicbackgroundinlinearalgebraandanalysisisassumed.Wedonotrequirefamiliaritywithcomplexnumbersandtherootsofunity.Theseareintroducedinthefirsttwosectionsofchapter1.Intheremainderofchapter1wereviewlinearalgebra.Students

5、shouldbefamiliarwiththebasicdefinitionsinsections1.3and1.4.Fromourviewpoint,lineartransformationsaretheprimaryobjectofstudy;vviPrefaceamatrixarisesasarealizationofalineartransformation.Manystudentsmayhavebeenexposedtothematerialonchangeofbasisinsectio

6、n1.4,butmaybenefitfromseeingitagain.Insection1.5,weaskhowtopickabasistosimplifythematrixrepresentationofagivenlineartransformationasmuchaspossible.Wethenfocusonthesimplestcase,whenthelineartransformationisdiagonalizable.Insection1.6,wediscussinnerprod

7、uctsandorthonormalbases.Weendwithastatementofthespectraltheoremformatrices,whoseproofisoutlinedintheexercises.Thisisbeyondtheexperienceofmostundergraduates.Chapter1isintendedasreferencematerial.Dependingonbackground,manyreadersandinstructorswillbeabl

8、etoskiporquicklyreviewmuchofthismaterial.Thetreatmentinchapter1isrelativelythorough,however,tomakethetextasself-containedaspossible,providealogicallyorderedcontextforthesubjectmatter,andmotivatelaterdevelopments.Theauthorbelievesthatstudentsshouldbei