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arXiv:math/0401271v1[math.CA]21Jan2004ARiemann-HilbertApproachtotheAkhiezerPolynomialsYangChen†,††DepartmentofMathematicsUniversityofWisconsin-Madison480LincolnDriveMadison,WI53706,USAAlexanderR.Its∗DepartmentofMathematicalSciencesIndianaUniversityPurdueUniversityIndiana402NorthBlackfordStreetIndianapolis,IN46202-3216,USA01/09/2003AbstractInthispaper,westudythosepolynomials,orthogonalwithrespecttoaparticularweight,overtheunioinofdisjointintervals,firstintroducedbyN.I.Akhiezer,viaareformulationasamatrixfactorizationorRiemann-Hilbertproblem.Thisapproachcomplementsthemethodproposedinapreviouspaper,thatinvolvestheconstructionofacertainmeromorphicfunctiononahyperellipticRiemannsurface.Themethodde-scribedhereisbasedonthegeneralRiemann-Hilbertschemeofthetheoryofintegrablesystemsandwillenableustoderive,inaverystrightforwardway,therelevantsystemofFuchsiandifferentialequationsforthepolynomialsandtheassociatedsystemoftheSchlesingerdeformationequationsforcertainquantatiesinvolvingthecorrespondingrecurrencecoefficients.BothoftheseequationswereobtainedearlierbyA.Magnus.Inourapproach,however,weareabletogobeyondMagnus’sresultsbyactuallysolvingtheequationsintermsoftheRiemannΘ-functions.WealsoshowthattherelatedHankeldeterminantcanbeinterpretedastherelevantτ−function.†ychen@ic.ac.uk,††Addressasof01/01/03:DepartmentofMathematics,ImperialCollege,180Queen’sGates,LondonSW72BZ,UK.∗itsa@math.iupui.edu1AcknowledgmentThefirstauthorshouldliketothanktheofDepartmentofMathematics,UniversityofWisconsin-Madison,forthekindhospitalityinhostinghimandtheEPSRCforaOver-seaTravelGrantthatmadethisendeavourpossible.ThesecondauthorwassupportedinpartbyNSFGrantDMS-0099812andbyImperialCollegeoftheUniversityofLondonviaaEPSRCGrant.ThefinalpartofthisprojectwasdonewhenhewasvisitingInstitutdeMath´ematiquedel’Universit´edeBourgogne,andthesupportduringhisstaythereisgratefullyacknowledged.1IntroductionTheChebyshevpolynomialsarethosemonicpolynomialscharacterisedbythepropertythatmax|πn(x)|,x∈[−1,1],isassmallaspossible.Indeed,itisalsoknownthatπnisorthog-onalwithrespectto1π√1−x2over[−1,1].Thepolynomialsπn—theChebyshevpolynomialsofthefirstkind—whichsatisfyaconstantcoefficientsthreetermrecurrencerelations,canbethoughtofasthe“HydrogenAtom”modelofthosepolynomialsorthogonalover[−1,1].TheseplayafundamentalroleinthelargenasymptoticsoftheBernstein-Szeg¨opolynomi-alswhichareorthogonalwithrespecta“deformed”Chebyshevweight,p(x)/√1−x2,over[−1,1],wherep(x)isstrictlypositive,absolutelycontinuousandsatisfiestheSzeg¨ocondition[21]Z1−1lnp(x)√1−x2dx−∞.ManyyearsagoN.I.AkhiezerandalsoYu.Ya.Tomchuk[1],[2],[3]consideredagen-eralizationoftheChebyshevpolynomials,wheretheintervaloforthogonalityisaunionofdisjointintervalshenceforthdenotedasE:=(β0,α1)∪(β1,α2)∪···∪(βg,βg+1).(1.1)ForcomparisonwiththoseofAkhiezer,weassumehereβ0=−1,andβg+1=1.Forlaterconvenience,whentheendpointsbecomeindependentvariablesweshalladopttheconven-tion,(α1,α2,...,αg,β0,β1,...,βg+1)−→(δ1,δ2,...,δg+1,δg+2,...,δ2g+2).(1.2)Letw(z):=iπsΠgj=1(z−αj)Πg+1j=0(z−βj),(1.3)bedefinedintheCP1\E.Themulti-intervalanalogoftheChebyshevweightisw+(t)=1πsΠgj=1(t−αj)(βg+1−t)(t−β0)Πgj=1(t−βj)0,t∈E,(1.4)2andisobtainedfromthecontinuationw(z)tothetopofthecut,E.ThegeneralizedCheby-shevorAkhiezerpolynomialsPnaremonicpolynomialsorthogonalwithrespecttow+,i.e.,ZEPm(x)Pn(x)w+(x)dx=hnδm,n,(1.5)wherehnisthesquareoftheL2norm.IntheconstructionoftheBernstein-Szeg¨oasymptoticsoverE,forpolynomialsorthogonalwithrespecttotheweightp(t)w+(t),wherepisanabsolutelycontinuouspositivefunction,exactinformationonPnwouldberequired.Thiswouldentailthesolutionofthe“HydrogenAtom”probleminthemultipleintervalsituation.Inthecaseoftwointervals,[−1,α]∪[β,1],PnwasconstructedbyAkhiezerwithaninnovationwhichwewouldnowrecogniseastheBaker-Akhiezerfunction,associatedwiththediscreteSchr¨odingerequation,namely,thethreetermrecurrencerelations,wherethedegreeofthepolynomialsnisthe“coordinates”,andzisspectralvariable.Akhiezer,basedhisconstructionontheconformalmappingofadoublyconnecteddomain,withtheaidoftheJacobianellipticfunctions,asademonstrationforhisstudents,theapplicationsofellipticfunctions[4].Itisnotatallclearhowtheconformalmappingcouldbeadeptedtohandlethesituationwhentherearemorethentwointervals.Intheearly1960’s,AkhiezerandalsowithTomchukpublishedseveralveryshortandverydeeppapersregardingtheBernstein-Szeg¨oasymptotics.AkhiezerandTomchukgaveadescriptionofPnandQn(thesecondsolutionoftherecurrencerelations)withtheaidoftheoryofHyperellipticintegralsintermsofacerianAbelianintegralofthethirdkind.However,certainunknownpointsonRiemannsurfaceappearinthisrespresentation,latercircumventedin[5].InarecentworkofA.P.Magnus[6],ageneralclassofsemi-classicalorthogonalpoly-nomials,whichincludestheAkhiezerpolynomialsPn,wasintroducedandshownthatthesepolynomialssatisfyacertainsystemoflinearFuchsianequations.Itwasalsodemonstratedtherethattherecurrencecoefficients,asfunctionsofthenaturalparametersofthesemi-classicalweights,obeythenonlinearSchlesingerequations,i.e.thedifferentialequationsdescribingtheisomonodromydeformat
本文标题:A Riemann-Hilbert Approach to the Akhiezer Polynom
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