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1、TheConstructionandApplicationofWaveletsinNumericalAnalysisWimSweldensMay18,1995iTheConstructionandApplicationofWaveletsinNumericalAnalysisAbstractWaveletsareanewfamilyofbasisfunctionsthatcanbeusedtoapproximategeneralfunctions.Theycombinepowerfulpropertiessuchas(bi)orthogonal
2、ity,compactsupport,localizationintimeandfrequency,andfastalgorithms.Thisthesisinvestigatestheuseofwaveletsinnumericalanalysisproblems.Inthe rstpartweconstructtwobasictools,quadratureformulaeandasymptoticerrorexpansions.Theformerprovidesaneasywaytocalculatethewaveletcoef- cie
3、nts,whilethelatterallowsasimplecomparisonofdi erentwaveletfamilies.Inthesecondpart,weconstructandstudywaveletsadaptedtoaweightedinnerproduct.Weshowhowonecanusethosewaveletsfortherapidsolutionofordinarydi erentialequations.Finally,westudysmoothlocaltrigonometricfunctions,whic
4、hcanbeseenastheFouriertransformofwavelets.Wegeneralizetheirconstructiontothebiorthogonalcase,andshowhowtousethemindatacompressionalgorithms.Thisisillustratedwithexamplesconcerningimagecompression.Keywords.wavelet,multiresolutionanalysis,quadratureformula,errorexpan-sion,weig
5、htedinnerproduct,ordinarydi erentialequations,localtrigonometricfunctions,compression.AMSsubjectclassi cation.41A30,42A10,42C05,65B05,65D32,65L60.iiAcknowledgementsThisthesiswouldneverhavebeenwrittenwithoutthehelpandsupportofnumer-ouspeople.NeedlesstosayIthankallofthemandwil
6、lalwaysbeartheire ortsinmyheart.Therefore,Iwouldliketoincludethefollowingshortstory.WhenIstartedasadoctoralstudent,nowmorethanthreeandahalfyearsago,Prof.RobertPiessensagreedtobemyadvisor;eversince,hehasbeensupportiveofmyactions.Hehasallowedmewidefreedominmychoiceofresearchar
7、eaandlocationatwhichtostudyit.Withhishelp,Iwasabletoexpandmyhorizonbyfrequentlyattendinginternationalconferences.Ijoinedtheparallelgroep",aresearchgroupfocusedonhighperformancescienti ccomputing.Threeofitsmembers,Prof.DirkRoose,DannyGovaerts,andStefanVandewalle,werethe rstt
8、odrawmyattentiontowavelets.Thiswasthestartingpointoftheresearchthatleadtoth
