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Re nableSplineFuntionsandHermiteInterpolationTimN.T.GoodmanAbstrat.Weonsiderr-vetorsofsplinefuntionswithompatsup-portandstableintegertranslateswhihsatisfyare nementequationwitha nitenumberofterms.Itisseenthatforr=1or2,thehoieofsuhvetorsisverylimited.Howeverforr 3thereissuÆient exibility,evenforsimpleuniformknots,toallowthesatisfationoffurtherusefulproperties.Mentionismadeofbiorthogonalityandorthogonalityprop-erties.Moreoveritisshownthatforeahr 3suhvetorsofsplinesanbeonstrutedwhiharefundamentalforr-Hermiteinterpolationandhavearbitrarilyhighsmoothness.x1.IntrodutionTaker 1andlet 1;:::; rbereal-valuedfuntionsonIRandwrite =( 1;:::; r)T.Wesaythat isre nableifitsatis esthere nementequation (x)=1Xj= 1Aj (2x j);x2IR;(1)whereAj;j2ZZ,arer rmatries.Muhworkhasbeendoneinstudyingonditionsonthematries(Aj)thatwillensurethat(1)hasauniquesolution satisfyingertainproperties,e.g.smoothness,polynomialreprodutionororthogonality.Thesolution isthende nedimpliitlyfromthere nementequation.Weheretakethedi erentapproahofstudyingonditionsonsplinefuntions,i.e.pieewisepolynomials,whihensurethattheysatisfyare nementequationofform(1).Inthisaseweknowexpliitlythesmoothnessandpolynomialreprodutionof ,butthevaluesofthematries(Aj)willbede nedimpliitlyfrom(1)andwillnotdiretlyonernus.Weshallkeepmatterssimple.Heneweassumethatonlya nitenumberofthematries(Aj)in(1)arenon-zero.Moreoverwesupposethat 1;:::; rhaveompatsupportandtheirintegertranslatesformaRieszbasis,i.e.forsomeonstantsA;B0,whenf=rXi=11Xj= 1aij i(: j);foranyaij2IR;i=1;:::;r;j2ZZ;2T.N.T.GoodmanwehaveArXi=11Xj= 1a2ij Z1 1f2 BrXi=11Xj= 1a2ij:(2)Forsimpliityweshallsay isstablewhentheaboveondition(2)holds.Inthere nementequation(1)itwouldbepossibletoreplaethedilationfator2byanyintegerm 2,butforsimpliityweonsideronlym=2.InSetion2weshallreviewsomegeneralresults,whihshowinpartiularthatforr=1and2thereislittle exibilityinthehoieofre nablesplinefuntionsasabove.However,forr 3thereisinontrastagreatdealof exibility,andthisanbeusedtoonstrutre nablesplinefuntionswithpropertieswhihareusefulforthetwomainappliationsofre nablefuntions.The rstoftheseonernswavelets,forwhihitisusualtorequiresomeorthogonalitypropertiesofthere nablefuntions.AttheendofSetion2wereallbrie ytheonstrutionforr=3ofre nablesplinefuntionswhiharebiorthogonaltoB-splineswithsimpleuniformknots.TheB-splinesherehavearbitrarydegreen 1andsimpleknotsat13ZZ,whiletheirdualfuntionshavethesamedegreeandsimpleknotsin14ZZ.Wealsomentiontheonstrutionoforthogonalre nablesplinefuntions,againforr=3,arbitrarydegreen 1andsimpleknotsin14ZZ,thedetailsofwhihwillappearinalaterpaper.Theothermainappliationofre nablefuntionsonernssubdivisionshemes.InSetion3,whihisallnewwork,weonstrutre nablesplinefuntions whiharefundamentalforr-Hermiteinterpolation,i.e.wehave (j 1)i(k)=ÆijÆko;i;j=1;:::;r;k2ZZ:(3)Weshowthatforeahr 3,itispossibletoonstrutsuhfundamentalre nablesplinesforarbitrarydegreen r+1andsimpleuniformknots,thusgivingsmoothnessCn 1.ThesegiverisetoHermiteinterpolatorysubdivisionshemes,i.e.shemesinwhihwearegiventhevaluesofafuntionanditsderivativesuptoorderr 1attheintegersandthesubdivisionsuessively llsinthevaluesofthefuntionanditsderivativesuptoorderr 1atthedyadipoints2 jk;k2ZZ;j=1;2;:::,oftheuniquere nablefuntionwhihinterpolatesthedata.Asfarasweknow,thesearetheonlyinterpolatorysubdivisionshemeswhih,forr 2,areknowntogivearbitrarilyhighsmoothnessoftheinterpolatingfuntions.x2.SomeGeneralitiesFortheaser=1ofasinglere nablesplinefuntion,itwasshownbyLawton,LeeandShenthatthehoieisveryrestritedindeed.Theorem1.[10℄Afuntion isastablere nablesplinefuntion(ofompatsupportsatisfyingare nementequationwitha nitenumberofterms)ifandonlyif isaonstantmultipleofaB-splinewithsimpleknotsattheintegers.Beforeanalysingtheaser=2,weonsidersomegeneralresults.Wesayavetor =( 1;:::; r)TgeneratesaspaeSifSomprisesall niteRe nableSplines3linearombinationsofintegertranslatesofelementsof .Thus1;:::;rlieinSifandonlyiffor=(1;:::;r)T,=rXi=11Xk= 1Ak (: k);(4)fora niteolletion(Ak)ofr rmatries.Equation(4)anbeexpressedmoreneatlybyintroduingtheFouriertransform^f(u):=Z1 1e iuxf(x)dx;u2IR:ThentakingFouriertransformsof(4)gives^(u)=A(e iu)^ (u);u2IR;(5)whereAdenotesther rmatrixofLaurentpolynomialsA(z):=1Xk= 1Akzk;z=e iu;u2IR:(6)Weshallsayar rmatrixAofLaurentpolynomials,asin(6),isinvertibleifithasaninverseA 1whihisalsoamatrixofLaurentpolynomials.ItiseasilyseenthatAisinvertibleifandonlyifitsdeterminantdetAisanon-trivialmonomial.TheFouriertransformalsogivesanelegantonditionforthestabilityof .ItisshownbyJiaandMihelli[9℄that =( 1;:::; r)TisstableifandonlyifforeahuinIR,thereareintegersk1;:::;krwithdet[^ i(u+2 kj)℄ri;j=16=0:(7)Fromtheabovedisussionweaneasilydeduethefollowing.Lemma2.Suppose isstableandgeneratesS.ThenisstableandalsogeneratesSifandonlyif^(u)=A(e iu)^ (u);u2IR;foraninvertiblematrixA.Weshallall andequivalentiftheysatisfytheonditionsofLemma2.Clearlyif andareequivalent,thentheyhavethesamenumberofompo-nents.Nowsupposethat isre nableandgeneratesS.ThenanyelementofSisa nitelinearombinationofintegertranslatesoftheelemen