The compact supp ort automatically ensures that is strictly ositiv de57356nite Another observ ation as that compactly supp orted radial functions can strictly ositiv de57356nite on only for 57356xed max imal v alue It is not ossible for function to ID: 69314
Download Pdf The PPT/PDF document "Chapter Compactly Supp orted Radial Basi..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Chapter4CompactlySupportedRadialBasisFunctionsAswesawearlier,compactlysupportedfunctionsthataretrulystrictlycondition-allypositivede niteoforderm�0donotexist.Thecompactsupportautomaticallyensuresthatisstrictlypositivede nite.Anotherobservationwasthatcompactlysupportedradialfunctionscanbestrictlypositivede niteonIRsonlyfora xedmax-imals-value.Itisnotpossibleforafunctiontobestrictlypositivede niteandradialonIRsforallsandalsohaveacompactsupport.Thereforewefocusourattentiononthecharacterizationandconstructionoffunctionsthatarecompactlysupported,strictlypositivede niteandradialonIRsforsome xeds.Accordingtoourearlierwork(Bochner'sTheoremandgeneralizationsthereof),afunctionisstrictlypositivede niteandradialonIRsifitss-variateFouriertransformisnon-negative.Theorem2.1.2givestheFouriertransformof='(kk)as^(x)=Fs'(r)=r (s 2)=2Z10'(t)ts=2J(s 2)=2(rt)dt:4.1OperatorsforRadialFunctionsandDimensionWalksSchabackandWu[564]de nedanintegraloperatoranditsinversedi erentialoperator,anddiscussedanentirecalculusforhowtheseoperatorsactonradialfunctions.Theseoperatorswillfacilitatetheconstructionofcompactlysupportedradialfunctions.De nition4.1.11.Let'besuchthatt7!t'(t)2L1[0;1),thenwede ne(I')(r)=Z1rt'(t)dt;r0:2.Foreven'2C2(IR)wede ne(D')(r)= 1r'0(r);r0:Inbothcasestheresultingfunctionsaretobeinterpretedasevenfunctionsusingevenextension.37 Remark:NotethattheoperatorIdi ersfromtheoperatorIintroducedearlierbyafactortintheintegrand.However,thetwooperatorsarerelated.Infact,wehaveI'(2=2)=I'(),i.e.,Z1rt'(t2=2)dt=Z1r2=2'(t)dt:Themostimportantpropertiesoftheseoperatorsare(see,e.g.,[564]or[627]):Theorem4.1.21.BothDandIpreservecompactsupport,i.e.,if'hascompactsupport,thensodoD'andI'.2.If'2C(IR)andt7!t (t)2L1[0;1),thenDI'='.3.If'2C2(IR)isevenand'02L1[0;1),thenID'='.4.Ift7!ts 1'(t)2L1[0;1)ands3,thenFs(')=Fs 2(I').5.If'2C2(IR)isevenandt7!ts'0(t)2L1[0;1),thenFs(')=Fs+2(D').TheoperatorsIandDallowustoexpresss-variateFouriertransformsas(s 2)-or(s+2)-variateFouriertransforms,respectively.Inparticular,adirectconsequenceoftheabovepropertiesandthecharacterizationofstrictlypositivede niteradialfunc-tions(Theorem2.4.1)isTheorem4.1.31.Suppose'2C(IR).Ift7!ts 1'(t)2L1[0;1)ands3,then'isstrictlypositivede niteandradialonIRsifandonlyifI'isstrictlypositivede niteandradialonIRs 2.2.If'2C2(IR)isevenandt7!ts'0(t)2L1[0;1),then'isstrictlypositivede niteandradialonIRsifandonlyifD'isstrictlypositivede niteandradialonIRs+2.Thisallowsustoconstructnewstrictlypositivede niteradialfunctionsfromgivenonesbya\dimension-walk"techniquethatstepsthroughmultivariateEuclideanspaceinevenincrements.4.2Wendland'sCompactlySupportedFunctionsIn[627]Wendlandconstructedapopularfamilyofcompactlysupportedradialfunctionsbystartingwiththetruncatedpowerfunction(whichweknowtobestrictlypositivede niteandradialonIRsfors2` 1),andthenwalkingthroughdimensionsbyrepeatedlyapplyingtheoperatorI.De nition4.2.1With'`(r)=(1 r)`wede ne's;k=Ik'bs=2c+k+1:Itturnsoutthatthefunctions's;kareallsupportedon[0;1]andhaveapolynomialrepresentationthere.Moreprecisely,38 Theorem4.2.2Thefunctions's;karestrictlypositivede niteandradialonIRsandareoftheform's;k(r)=ps;k(r);r2[0;1];0;r�1;withaunivariatepolynomialps;kofdegreebs=2c+3k+1.Moreover,'s;k2C2k(IR)areuniqueuptoaconstantfactor,andthepolynomialdegreeisminimalforgivenspacedimensionsandsmoothness2k.Wendlandgaverecursiveformulasforthefunctions's;kforalls;k.Weinsteadlisttheexplicitformulasof[195]Theorem4.2.3Thefunctions's;k,k=0;1;2;3,havetheform's;0(r)=(1 r)`;'s;1(r):=(1 r)`+1+[(`+1)r+1];'s;2(r):=(1 r)`+2+(`2+4`+3)r2+(3`+6)r+3;'s;3(r):=(1 r)`+3+(`3+9`2+23`+15)r3+(6`2+36`+45)r2+(15`+45)r+15;where`=bs=2c+k+1,andthesymbol:=denotesequalityuptoamultiplicativepositiveconstant.Proof:Thecasek=0followsdirectlyfromthede nition.Applicationofthede nitionforthecasek=1yields's;1(r)=(I'`)(r)=Z1rt'`(t)dt=Z1rt(1 t)`+dt=Z1rt(1 t)`dt=1(`+1)(`+2)(1 r)`+1[(`+1)r+1];wherethecompactsupportof'`reducestheimproperintegraltoade niteintegralwhichcanbeevaluatedusingintegrationbyparts.TheothertwocasesareobtainedsimilarlybyrepeatedapplicationofI.Examples:Fors=3wegetsomeofthemostcommonlyusedfunctionsas'3;0(r)=(1 r)2;2C0\SPD(IR3)'3;1(r):=(1 r)4(4r+1);2C2\SPD(IR3)'3;2(r):=(1 r)6 35r2+18r+3;2C4\SPD(IR3)'3;3(r):=(1 r)8 32r3+25r2+8r+1;2C6\SPD(IR3):39 4.3Wu'sCompactlySupportedFunctionsIn[656]Wupresentsanotherwaytoconstructstrictlypositivede niteradialfunctionswithcompactsupport.Hestartswiththefunction (r)=(1 r2)`;`2IN;whichisstrictlypositivede niteandradialsinceweknowthatthetruncatedpowerfunction (p)ismultiplymonotone.Wuthenconstructsanotherfunctionthatisstrictlypositivede niteandradialonIRbyconvolution,i.e., `(r)=( )(2r)=Z1 1(1 t2)`(1 (2r t)2)`+dt=Z1 1(1 t2)`(1 (2r t)2)`dt:Thisfunctionisstrictlypositivede nitesinceitsFouriertransformisessentiallythesquareoftheFouriertransformof .JustliketheWendlandfunctions,thisfunctionisapolynomialonitssupport.Infact,thedegreeofthepolynomialis4`+1,and `2C2`(IR).Now,afamilyofstrictlypositivede niteradialfunctionsisconstructedbyadi-mensionwalkusingtheDoperator,i.e., k;`=Dk `:Thefunctions k;`arestrictlypositivede niteandradialinIRsfors2k+1,arepolynomialsofdegree4` 2k+1ontheirsupportandinC2(` k)intheinteriorofthesupport.OntheboundarythesmoothnessincreasestoC2` k.Example:For`=3wecancomputethethreefunctions k;3(r)=Dk 3(r)=Dk((1 2)3(1 2)3)(2r);k=0;1;2;3:Thisresultsin 0;3(r):= 5 39r2+143r4 429r6+429r7 143r9+39r11 5r13+:=(1 r)7(5+35r+101r2+147r3+101r4+35r5+5r6)2C6\SPD(IR) 1;3(r):= 6 44r2+198r4 231r5+99r7 33r9+5r11+:=(1 r)6(6+36r+82r2+72r3+30r4+5r5)2C4\SPD(IR3) 2;3(r):= 8 72r2+105r3 63r5+27r7 5r9+:=(1 r)5(8+40r+48r2+25r3+5r4)2C2\SPD(IR5) 3;3(r):= 16 35r+35r3 21r5+5r7+:=(1 r)4(16+29r+20r2+5r3)2C0\SPD(IR7):Remarks:1.ForaprescribedsmoothnessthepolynomialdegreeofWendland'sfunctionsislowerthanthatofWu'sfunctions.Forexample,bothWendland'sfunction'3;2andWu'sfunction 1;3areC4smoothandstrictlypositivede niteandradialinIR3.However,thepolynomialdegreeofWendland'sfunctionis8,whereasthatofWu'sfunctionis11.40 10.40.8r0.51-1-0.50010.60.80r10-10.20.5-0.510.80.600.2r10.50-1-0.5Figure4.1:PlotofWendland'sfunctions(left),Wu'sfunctions(center),andBuhmann'sfunction(right)listedasexamples.2.Whilebothfamiliesofstrictlypositivede nitecompactlysupportedfunctionsareconstructedviadimensionwalk,Wendlandusesintegration(andthusobtainsafamilyofincreasinglysmootherfunctions),whereasWuneedstostartwithafunctionofsucientsmoothness,andthenobtainssuccessivelylesssmoothfunctions(viadi erentiation).4.4Buhmann'sCompactlySupportedFunctionsAthirdfamilyofcompactlysupportedstrictlypositivede niteradialfunctionsthathasappearedintheliteratureisduetoBuhmann(see[84]).Buhmann'sfunctionscontainalogarithmicterminadditiontoapolynomial.Hisfunctionshavethegeneralform (r)=Z10(1 r2=t)t (1 t)dt:Here012,1,andinordertoobtainfunctionsthatarestrictlypositivede niteandradialonIRsfors3theconstraintsfortheremainingparametersare0,and 1 12.Example:Anexamplewith ==12,=1and=2islistedin[85]: (r):=12r4logr 21r4+32r3 12r2+1;0r1;2C2\SPD(IR3):Remarks:1.WhileBuhmann[85]claimsthathisconstructionencompassesbothWendland'sandWu'sfunctions,Wendland[634]givesanevenmoregeneraltheoremthatshowsthatintegrationofapositivefunctionf2L1[0;1)againstastrictlyposi-tivede nite(compactlysupported)kernelKresultsina(compactlysupported)strictlypositivede nitefunction,i.e.,'(r)=Z10K(t;r)f(t)dt41 isstrictlypositivede nite.Buhmann'sconstructionthencorrespondstochoosingf(t)=t (1 t)andK(t;r)=(1 r2=t).2.MultiplymonotonefunctionsarecoveredbythisgeneraltheorembytakingK(t;r)=(1 rt)k 1+andfanarbitrarypositivefunctioninL1sothatd(t)=f(t)dtinWilliamson'scharacterizationTheorem2.6.2.Also,functionsthatarestrictlypositivede niteandradialinIRsforalls(orequivalentlycompletelymonotonefunctions)arecoveredbychoosingK(t;r)=e rt.42