an introduction to wavelets through linear algebra

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1、PrefaceMathematicsmajorsatMichiganStateUniversitytakea“Capstone”courseneartheendoftheirundergraduatecareers.Thecontentofthiscoursevarieswitheachoffering.Itspurposeistobringtogetherdifferenttopicsfromtheundergraduatecurriculumandintroducestudentstoadeveloping

2、areainmathematics.ThistextwasoriginallywrittenforaCapstonecourse.Basicwavelettheoryisanaturaltopicforsuchacourse.Byname,waveletsdatebackonlytothe1980s.Ontheboundarybetweenmathematicsandengineering,wavelettheoryshowsstudentsthatmathematicsresearchisstillthriv

3、ing,withimportantapplicationsinareassuchasimagecompressionandthenumericalsolutionofdifferentialequations.Theauthorbelievesthattheessentialsofwavelettheoryaresufficientlyelementarytobetaughtsuccessfullytoadvancedundergraduates.Thistextisintendedforundergraduat

4、es,soonlyabasicbackgroundinlinearalgebraandanalysisisassumed.Wedonotrequirefamiliaritywithcomplexnumbersandtherootsofunity.Theseareintroducedinthefirsttwosectionsofchapter1.Intheremainderofchapter1wereviewlinearalgebra.Studentsshouldbefamiliarwiththebasicdefin

5、itionsinsections1.3and1.4.Fromourviewpoint,lineartransformationsaretheprimaryobjectofstudy;vviPrefaceamatrixarisesasarealizationofalineartransformation.Manystudentsmayhavebeenexposedtothematerialonchangeofbasisinsection1.4,butmaybenefitfromseeingitagain.Insec

6、tion1.5,weaskhowtopickabasistosimplifythematrixrepresentationofagivenlineartransformationasmuchaspossible.Wethenfocusonthesimplestcase,whenthelineartransformationisdiagonalizable.Insection1.6,wediscussinnerproductsandorthonormalbases.Weendwithastatementofthe

7、spectraltheoremformatrices,whoseproofisoutlinedintheexercises.Thisisbeyondtheexperienceofmostundergraduates.Chapter1isintendedasreferencematerial.Dependingonbackground,manyreadersandinstructorswillbeabletoskiporquicklyreviewmuchofthismaterial.Thetreatmentinc

8、hapter1isrelativelythorough,however,tomakethetextasself-containedaspossible,providealogicallyorderedcontextforthesubjectmatter,andmotivatelaterdevelopments.Theauthorbelievesthatstudentsshouldbei