Orthogonal polynomials and cubature formulae on sp

ORTHOGONALPOLYNOMIALSANDCUBATUREFORMULAEONSPHERESANDONBALLSYUANXUySIAMJ.MATH.ANAL.c1998SocietyforIndustrialandAppliedMathematicsVol.29,No.3,pp.779{793,May1998015Abstract.OrthogonalpolynomialsontheunitsphereinRd+1andontheunitballinRdareshowntobecloselyrelatedtoeachotherforsymmetricweightfunctions.Furthermore,itisshownthatalargeclassofcubatureformulaeontheunitspherecanbederivedfromthoseontheunitballandviceversa.Theresultsprovideanewapproachtostudyorthogonalpolynomialsandcubatureformulaeonspheres.Keywords.orthogonalpolynomialsinseveralvariables,onspheres,onballs,sphericalhar-monics,cubatureformulaeAMSsubjectclassications.33C50,33C55,65D32PII.S00361410963073571.Introduction.Weareinterestedinorthogonalpolynomialsinseveralvari-ableswithemphasisonthoseorthogonalwithrespecttoagivenmeasureontheunitsphereSdinRd+1.IncontrasttoorthogonalpolynomialswithrespecttomeasuresdenedontheunitballBdinRd,therehavebeenrelativelyfewstudiesonthestruc-tureoforthogonalpolynomialsonSdbeyondtheordinarysphericalharmonicswhichareorthogonalwithrespecttothesurface(Lebesgue)measure(cf.[2,3,4,5,6,8]).TheclassicaltheoryofsphericalharmonicsisprimarilybasedonthefactthattheordinaryharmonicssatisfytheLaplaceequation.RecentlyDunkl(cf.[2,3,4,5]andthereferencestherein)openedawaytostudyorthogonalpolynomialsonthesphereswithrespecttomeasuresinvariantunderanitereectiongroupbydevelopingatheoryofsphericalharmonicsanalogoustotheclassicalone.Inthisimportantthe-orytheroleofLaplacianoperatorisreplacedbyadierential-dierenceoperatorinthecommutativealgebrageneratedbyafamilyofcommutingrst-orderdierential-dierenceoperators(Dunkl'soperators).Otherthantheseresults,however,wearenotawareofanyothermethodofstudyingorthogonalpolynomialsonspheres.Acloselyrelatedquestionisconstructingcubatureformulaeonspheresandonballs.Cubatureformulaewithaminimalnumberofnodesareknowntoberelatedtoorthogonalpolynomials.Overtheyears,alotofeorthasbeenputintothestudyofcubatureformulaeformeasuressupportedontheunitball,oronothergeometricdomainswithnonemptyinteriorinRd.Incontrast,thestudyofcubatureformulaeontheunitspherehasbeenmoreorlessfocusedonthesurfacemeasureonthesphere;thereislittleworkontheconstructionofcubatureformulaewithrespecttoothermeasures.Thisispartlyduetotheimportanceofcubatureformulaewithrespecttothesurfacemeasure,whichplayaroleinseveraleldsinmathematics,andperhapspartlyduetothelackofstudyoforthogonalpolynomialswithrespecttoageneralmeasureonthesphere.OnemainpurposeofthispaperistoprovideanelementaryapproachtowardsthestudyoforthogonalpolynomialsonSdforalargeclassofmeasures.ThisapproachisReceivedbytheeditorsJuly26,1996;acceptedforpublication(inrevisedform)January9,1997.ThisresearchwassupportedbytheNationalScienceFoundationundergrantDMS-9500532.([e-mail]).779780YUANXUbasedonacloseconnectionbetweenorthogonalpolynomialsonSdandthoseontheunitballBd;aprototypeoftheconnectionisthefollowingelementaryexample.Ford=1,thesphericalharmonicsofdegreenaregiveninthestandardpolarcoordinatesbyY(1)n(x1;x2)=rncosnandY(2)n(x1;x2)=rnsinn:(1.1)Underthetransformx=cos,thepolynomialsTn(x)=cosnandUn(x)=sinn=sinaretheChebyshevpolynomialsoftherstandthesecondkind,orthogonalwithre-spectto1=p1x2andp1x2,respectively,ontheunitball[1;1]inR.Hence,thesphericalharmonicsonS1canbederivedfromorthogonalpolynomialsonB1.WeshallshowthatforalargeclassofweightfunctionsonRd+1wecancon-structhomogeneousorthogonalpolynomialsonSdfromthecorrespondingorthog-onalpolynomialsonBdinasimilarway.ThisallowsustoderivepropertiesoforthogonalpolynomialsonSdfromthoseonBd;thelatterhavebeenstudiedmuchmoreextensively.Althoughtheapproachiselementaryandthereisnodierentialordierential-dierenceoperatorinvolved,theresultoersanewwaytostudythestructureoforthogonalpolynomialsonSd.OurapproachdependsonanelementaryformulathatlinkstheintegrationonBdtotheintegrationonSd.ThesameformulayieldsanimportantconnectionbetweencubatureformulaeonSdandthoseonBd;theresultstatesroughlythatalargeclassofcubatureformulaeonSdisgeneratedbycubatureformulaeonBdandviceversa.Inparticular,itallowsustoshiftourattentionfromthestudyofcubatureformulaeontheunitspheretothestudyofcubatureformulaeontheunitball;therehasbeenmuchmoreunderstandingtowardsthestructureofthelatterone.Althoughtheresultissimpleandelementary,itsimportanceisapparent.Ityields,inparticular,manynewcubatureformulaeonspheresandonballs.Becausethemainfocusofthispaperisontherelationbetweenorthogonalpolynomialsandcubatureformulaeonspheresandthoseonballs,wewillpresentexamplesofcubatureformulaeinaseparatepaper.Thepaperisorganizedasfollows.Insection2weintroducenotationandpresentthenecessarypreliminaries,wherewealsoprovethebasiclemma.Insection3weshowhowtoconstructorthogonalpolynomialsonSdfromthoseonBd.Insection4wediscusstherelationbetweencubatureformulaeontheunitsphereandthoseontheunitball.2.Preliminaryandbasiclemma.Forx;y2Rdweletx
ydenotetheusualinnerproductofRdandjxj=(x
x)1=2theEuclideannor