On two families of near-best spline quasi-interpol

arXiv:math/0602305v1[math.NA]14Feb2006Ontwofamiliesofnear-bestsplinequasi-interpolantsonnon-uniformpartitionsofthereallineD.Barrera,M.J.Ib´a˜nez,P.Sablonni`ere,D.SbibihJanuary2006AbstractTheunivariatesplinequasi-interpolants(abbr.QIs)studiedinthispaperareapproximationoperatorsusingB-splineexpansionswithcoeffi-cientswhicharelinearcombinationsofdiscreteorweightedmeanvaluesofthefunctiontobeapproximated.Whenworkingwithnonuniformpar-titions,themainchallengeistofindQIswhichhavebothgoodapprox-imationordersanduniformnormswhichareboundedindependentlyofthegivenpartition.Near-bestQIsareobtainedbyminimizinganupperboundoftheinfinitynormofQIsdependingonacertainnumberoffreeparameters,thusreducingthisnorm.Thispaperisdevotedtothestudyoftwofamiliesofnear-bestQIsofapproximationorder3.Keywords:splineapproximation,splinequasi-interpolants.AMSclassification:41A15,41A35,65D07.1IntroductionAsplinequasi-interpolant(abbr.QI)offhasthegeneralformQf=Xα∈Aλα(f)Bαwhere{Bα,α∈A}isafamilyofB-splinesformingapartitionofunityand{λα,α∈A}isafamilyoflinearfunctionalswhicharelocalinthesensethattheyonlyusevaluesoffinsomeneighbourhoodofΣα=supp(Bα).ThemaininterestofQIsisthattheyprovidegoodapproximantsoffunctionswithoutsolvinganylinearsystemofequations.Inthispaper,wewanttostudythefollowingtypesofQIs:DiscreteQuasi-Interpolants(abbr.dQIs):thelinearfunctionalsarelinearcombinationsofvaluesoffatsomepointsinaneighbourhoodofΣα(seee.g.[1]-[3],[6]-[9],[11],[13][14][23][24]).1IntegralQuasi-Interpolants(abbr.iQIs):thelinearfunctionalsarelinearcom-binationsofweightedmeanvaluesoffinsomeneighbourhoodofΣα(seee.g.[2]-[5],[14],[23]-[26]).Morespecifically,westudyQIsthatwecallNear-BestQuasi-Interpolants(abbr.NBQIs)whicharedefinedasfollows:1)Near-BestdQIs:assumethatλα(f)=Pβ∈Fαaα(β)f(xβ)wherethefinitesetofpoints{xβ,β∈Fα}liesinsomeneighbourhoodofΣα.Thenitisclearthat,forkfk∞≤1andα∈A,|λα(f)|≤kaαk1,whereaαisthevectorwithcomponentsaα(β),fromwhichwededuceimmediatelykQk∞≤Xα∈A|λα(f)|Bα≤maxα∈A|λα(f)|≤maxα∈Akaαk1=ν1(Q).Now,assumingthatn=card(Fα)forallα,wecantrytofinda∗α∈Rnsolutionoftheminimizationproblemka∗αk1=min{kaαk1;aα∈Rn,Vαaα=bα}wherethelinearconstraintsexpressthatQisexactonsomesubspaceofpoly-nomials.Thus,wefinallyobtainkQk∞≤ν∗1(Q)=maxα∈Aka∗αk1.2)Near-BestiQIs:assumethatλα(f)=Pβ∈Fαaα(β)RΣβMβ(t)f(t)dt,wheretheB-splinesMβarenormalizedbyRMβ=1.NotethattheB-splineMβcanbedifferentfromBα.Onceagain,forkfk∞≤1,wehave|λα(f)|≤Xβ∈Fα|aα(β)||ZΣβMβ(t)f(t)dt|≤Xβ∈Fα|aα(β)|=kaαk1whence,asweobtainedabovefordQIs,kQk∞≤maxα∈Akaαk1=ν∗1(Q).AsemphasizedbydeBoor(seee.g.[4],chapterXII),aQIdefinedonnonuniformpartitionshastobeuniformlyboundedindependentlyofthepartitioninordertobeinterestingforapplications.Therefore,theaimofthispaperistodefinesomefamiliesofdiscreteandintegralQIssatisfyingthispropertyandhavingthesmallestpossiblenorm.Asingeneralitisdifficulttominimizethetruenormoftheoperator,wehavechosentosolvetheminimizationproblemsdefinedabove.Thepaperextendssomeresultsof[1][11],andisorganizedasfollows.Wefirstrecallsome”classical”QIsofvarioustypesandweverifythattheyareuniformlybounded.Thenwedefineandstudyseveralfamiliesofdiscreteand2integralQIs,dependingonafinitenumberofparameters,forwhichwecanfindν∗1(Q).Weshowthatthisproblemhasalwaysasolution(ingeneralnonunique).Ofparticularinterestaretheresultsoftheorems1,4,6and8whereweshowthatsomefamiliesofdQIsandiQIsareuniformlyboundedindependentlyofthepartition.Byimposingmoreconstraintsonthenon-uniformpartitions,wecanalsoprovethatsomefamiliesofQIsarenear-best(theorems5and9).(AparallelstudyofsplineQIsisdonein[2]foruniformpartitionsofthereallineandin[3]forauniformtriangulationoftheplane).Inallcases,theQIsthatwestudyareonlyexactonP2,i.e.theirapproximationorderis3.ItseemssurprisinglydifficulttoconstructQIswhicharebothuniformlyboundedindependentlyofthepartitionandexactonPdford≥3.Inthelastsection11,weconsideranexampleofQIwhichisexactonP3(i.e.ofapproximationorder4)anduniformlybounded:howeverthebounddependsonthemaximalmeshratioofthepartition.Suchoperatorsseemalsousefulforapplicationsanditwouldbeinterestingtostudynear-bestoperatorsofthistype,thusallowingareductionoftheupperboundofthenorm.2NotationsWeshalluseclassicalB-splinesofdegreemonaboundedintervalI=[a,b]oronI=R.Forthesakeofsimplicity,inthecaseI=R,wetakeastrictlyincreasingsequenceofknotsT={ti,i∈Z}satisfying|ti|→+∞as|i|→+∞.InthecaseI=[a,b],wetaketheusualsequenceTofknotsdefinedby(seee.g.[4],[9]):t−m=···=t0=at1t2···tn−1b=tn=···=tn+m.ForJ={0,...,n+m−1},thefamilyofB-splines{Bj,j∈J},withsupportΣj=[tj−m,tj+1]isabasisofthespaceSm(I,T)ofsplinesofdegreemontheintervalIendowedwiththepartitionT.TheseB-splinesformapartitionofunity,i.e.Pj∈JBj=1.Wesethi=ti−ti−1forallindicesi.LetNm={1,...,m}andTj={tj−r+1,r∈Nm}:werecallthattheelementarysymmetricfunctionsσl(T)ofthemvariablesinTjaredefinedbyσ0(Tj)=0andfor1≤l≤m,byσl(Tj)=X1≤r1r2...rl≤mtj+1−r1tj+1−r2...tj+1−rl.DenotingClm=m!l!(m−l)!thebinomialcoefficients,then,for0≤l≤m,themonomialsel(x)=xlcanbewrittenel=Pi∈Jθ(l)iBi,withθ(l)i=σl(Ti)/Clm.ThisisadirectconsequenceofMarsden’sidentity([4],chapterIX).Inparticular,theGrevi