Lagrange interpolation on Leja points

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Lagrange interpolation on Leja points

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Lagrange interpolation on Leja points
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Taylor, Rodney
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University of South Florida
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Equilibrium distribution
Fekete points
Lebesgue constants
Newton interpolation
Potential theory
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Dissertation (Ph.D.)--University of South Florida, 2008.
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Includes bibliographical references.
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by Rodney Taylor.

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Lagrange interpolation on Leja points
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Equilibrium distribution
Fekete points
Lebesgue constants
Newton interpolation
Potential theory
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LagrangeInterpolationonLejaPointsbyRodneyTaylorAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaCo-MajorProfessor:VilmosTotik,Sc.D.Co-MajorProfessor:BorisShekhtman,Ph.D.ArthurDanielyan,Ph.D.EvgueniiRakhmanov,Sc.D.DateofApproval:April1,2008Keywords:Equilibriumdistribution,Feketepoints,Lebesgueconstants,Newtoninterpolation,potentialtheorycCopyright2008,RodneyTaylor

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AcknowledgmentsIwishtothanktheDepartmentofMathematicsatTheUniversityofSouthFloridaforgivingmetheopportunityandthesupporttonishmydoctoratestudies.Iwouldalsoliketosincerelythankthemembersofmygraduatecommittee,Dr.Totik,Dr.Shehktman,Dr.Rakhmanov,andDr.Danyielanfortheirsupport.HoweverIthinkspecialthanksisduetoDr.Totik,forhisvaluablehelp,support,encouragement,andpatienceinlettingmendmyownway.Hetrulywantedthistobeadissertationtobeproudof.

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TableofContentsAbstractii1Introduction12Preliminaries32.1LagrangeInterpolation...........................32.2PotentialTheory..............................82.3LejaPoints.................................112.4EquilibriumMeasures............................123HierarchyofInterpolationSchemes233.1Proofofiii..............................243.2Proofofiiiii..............................263.3Proofofiv=ii.............................274LebesgueConstantsonLejaPointson[-1,1]304.1SeparationofLejaPoints..........................304.2ReductionoftheMainProof........................334.3EstimateforTheMainProduct......................355LejaPointsonMoreGeneralSets495.1LejaPointsonanArc...........................495.2LejaPointsonMoreGeneralSets.....................56References59AbouttheAuthorEndPagei

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LagrangeInterpolationonLejaPointsRodneyTaylorABSTRACTInthisdissertationweinvestigateLagrangeinterpolation.Ourrstresultwilldealwithahierarchyofinterpolationschemes.Specically,wewillshowthatgivenatriangulararrayofpointsinaregularcompactsetK,suchthatthecorrespondingLebesgueconstantsaresubexponential,onealwayshastheuniformconvergenceofLnftofforallfunctionsanalyticonK.WewillthenshowthatuniformconvergenceofLnftofforallanalyticfunctionsfisequivalenttothefactthattheprobabilitymeasuresn=1 nPnj=1zn;j,whichareassociatedwithourtriangulararray,convergeweakstartotheequilibriumdistributionforK.Motivatedbyourhierarchy,wewillthencometoourmainresult,namelythattheLebesgueconstantsassociatedwithLejasequencesonfairlygeneralcompactsetsaresubexponential.Moregenerally,consideringNewtoninterpolationonasequenceofpoints,wewillshowthattheweakstarconvergenceoftheircorrespondingprobabil-itymeasurestotheequilibriumdistribution,togetherwithacertaindistancingrule,impliesthattheircorrespondingLebesgueconstantsaresub-exponential.ii

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1IntroductionThefocusofthisdissertationisLagrangeinterpolation.Weheregiveabriefoverviewofourresults,withtechnicaldetailsomitted.Fromtheabundanceofliteratureonpolynomialapproximation,oneofthemostclassicalresults,Weierstrass'stheoremsee[11],page159,tellsusthateverycontin-uousfunctiononacompactsetKcanbeuniformlyapproximatedbypolynomials.WhenKisaniteclosedintervalontherealline,weknowthatgivenanf2CK,ifpnisthebestapproximanttofinthespacePn,thespaceofpolynomialsofdegreeatmostn,thentheequationpnx=fx.0.1hasn+1solutionsonK.Fromthisitfollowsthatifforeachnweinterpolateatthesolutionsto1.0.1,wewillhavekLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(fkK!0:.0.2Itisnottruehowever,that1.0.2mustholdforinterpolationonanarbitrarytri-angulararrayofpoints.InfactwhenK=[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1],thecelebratedtheoremofFabersee[7],page27claimsthatgivenanytriangulararrayofpointsasetofpointsfzn;kg1kn;n=1;2;:::,wherezn;j6=zn;kwheneverj6=kthereisalwaysafunctionf2CKsuchthatkLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(fkK90;andevenkLnfkK!1:.0.3Inlightof.0.3,thequestionbecomeswhataregoodpointsatwhichtointerpolate.Ourrstresult,ahierarchyofinterpolationschemes,ndsconditionsthatmakeatriangulararraygoodpointsatwhichtointerpolate.Specically,wendconditionssuchthat.0.2holdsforallfunctionsfwhichareanalyticonK.Wewillshowthat.0.2istrueifandonlyifcertainmeasuresassociatedwithourinterpolationpointsconvergeweakstartotheequilibriumdistributionforKwesayasequenceofmeasuresfngonacompactsetKconvergesweakstartoifRKftdnt!RKftdtforallf2CK.Asidefromhaving.0.2holdforalargeclassoffunctions,thereisanotherprop-ertywhichmakesanarrayofpointsdesirablefromaninterpolationstandpoint.IfpnisthebestapproximantinthespacePn,thesetofpolynomialsofdegreelessthanorequalton,toafunctionf,wehavekLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(fkKkpn)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fkKkLnk+1:1

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ItisthusdesirabletokeepthenormkLnkofLn,whichwecallthen-thLebesgueconstant,assmallaspossiblewhenwespeakofthenormofLn,wearespeakingofthestandardoperatornorm.Generallyspeaking,theLebesgueconstantsassociatedwithatriangulararrayofpointswillbesmallwhenthepointsarespreadout.Themeasureofsmallnesswhichweshalluseinthisdissertationissubexponentiality,thatis,wewillndinterpolationpointssuchthatkLnk1=n!1:DespitethefactthatpointswhicharesomewhatspreadoutwilltendtohavesmallLebesgueconstants,itisnottruethatforthearrayofequidistantnodestheLebesgueconstantsaresubexponentialsee[15].Itistrivialtoshowhowever,thattheLebesgueconstantsassociatedwithFeketesetssee[12],page142aresubexponential.AnnthFeketesetforacompactsetKisanysetofpointsz1;z2;:::;znwhichmaximizesVz1;:::;zn=Y1i
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2Preliminaries2.1LagrangeInterpolationAsstatedintheintroduction,theWeierstrasstheoremisnon-constructive,andthusthequestionbecomeshowtogenerategoodpolynomialapproximants.OnewaytogeneratepolynomialapproximantstoacontinuousfunctionisbyLagrangeinterpola-tion.Againtorepeatfromtheintroduction,thereexistsanimmenselyvastamountofliteratureonLagrangeinterpolation,seee.g.[13]andthereferencesthere.Weherepresentonlythebasicknowledgewhichweshallneedinthisdissertation.Lagrangeinterpolationcomeswhenweinterpolatetofunctionvalues,i.e.,whenwearegivenndistinctpointsinthecomplexplane,z1;z2;:::;zn,andwendtheuniquepolynomial,pn)]TJ/F25 7.97 Tf 6.587 0 Td[(1z,ofdegreen)]TJ/F19 11.955 Tf 11.955 0 Td[(1,suchthatpn)]TJ/F25 7.97 Tf 6.587 0 Td[(1zi=fzi;i=1;2;:::;n:.1.1Thatsuchapolynomialexistsiseasilyseenthroughthefunctionslkz=nYj=1;j6=kz)]TJ/F27 11.955 Tf 11.955 0 Td[(zj zk)]TJ/F27 11.955 Tf 11.955 0 Td[(zj;k=1;2;:::;n:.1.2Indeed,lkzj=j;k,anditisthuseasilyseenthatpn)]TJ/F25 7.97 Tf 6.586 0 Td[(1z=nXk=1fzklkz2.1.3isasolutionto2.1.1.Thatthissolutionisuniqueisalsoeasilyseen.Indeed,ifp1andp2arebothsolutionsto2.1.1,thenp=p1)]TJ/F27 11.955 Tf 10.663 0 Td[(p2isann)]TJ/F19 11.955 Tf 10.662 0 Td[(1thdegreepolynomialhavingnzeros,andthusitisthezeropolynomial.Ingeneral,whenonespeaksofLagrangeinterpolation,onespeaksofacompactsetKandatriangulararrayofpointswhichlieinK.Thatis,onespeaksofasetofpointsoftheformzj=zn;j,wherethereareexactlyndistinctpointsforeachn=1;2;::::z1;1z2;1z2;2z3;1z3;2z3;3.........3

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Associatedwiththistriangulararrayofpointsisthenasequence,fLng,ofLagrangeoperators,whereLnisdenedasin.1.3usingthenthrowofthetriangulararray,i.e.,Lnfz=Lnf;z=nXk=1fzn;kln;kz:.1.4Ifwewishtoemphasizethepointswhichweareinterpolatingat,weshallusethenotationLnf;z=Lnf;z1;:::;zn;z:AsanoperatorfromCKintoPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1,thesetofpolynomialsofdegreelessthanorequalton)]TJ/F19 11.955 Tf 11.955 0 Td[(1,Lnisgiventheusualnorm:kLnk=supkfk=1kLnfk;wherekk=kkKontherightisthesupremumnormonK.WecallthisnormthenthLebesgueconstantassociatedwiththearray.Itisclearfrom.1.4thatLnisalinearmap,andthatLnp=pforallp2Pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1.ThesetwopropertiesallowustousetheLebesgueconstantasarstestimateindetermininghowwellLnfapproximatesf.Ifpn)]TJ/F25 7.97 Tf 6.587 0 Td[(1isthebestapproximanttofinPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1,wehavethefollowingkLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(fk=kLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(pn)]TJ/F25 7.97 Tf 6.587 0 Td[(1+pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fk=.1.5kLnf)]TJ/F27 11.955 Tf 11.955 0 Td[(pn)]TJ/F25 7.97 Tf 6.587 0 Td[(1+pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fkkLnf)]TJ/F27 11.955 Tf 11.956 0 Td[(pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1k+kpn)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fkkLnkkpn)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fk+kpn)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fk=kpn)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(fkkLnk+1:Forgivenpoints,z1;z2;:::;zn,itiseasilyseenthatkLnk=supz2KnXk=1jlkzj:.1.6Indeed,kLnfk=supz2KnXk=1fzklkzkfksupz2KnXk=1jlkzj;.1.7fromwhereitfollowsthatkLnksupz2KnXk=1jlkzj:.1.8Butiftherighthandsideof.1.8attainsitsmaximumatz,thenforafunctionfwithkfk=1andfzk= lkz jlkzj;weactuallyhaveequalityin.1.7,therebygivingus2.1.6.4

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Itisclearfrom.1.5thatitisdesirabletochooseinterpolationnodessothattheLebesgueconstantsaresmall.Inspectionof.1.6and.1.2tellsusthattheLebesgueconstantswillbesmallwhenthenodesaresomewhatspreadout.Thenotionofsmallnesswhichweshalluseinthisdissertationwillbesubexponentiality,i.e.kLnk1=n!1.Equallyspacednodeswouldbetheeasiestselection,howevertheLebesgueconstantsassociatedwiththesenodesarenotsubexponentialsee[15].Ifhowever,thenthrowofatriangulararrayisannthFeketeset,itistrivialtoshowthattheLebesgueconstantsaresubexponential.RecallthatannthFeketesetforacompactsetKisanysystemofpointsmaximizingVz1;:::;zn=Y1i
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wherepnt=Qni=1t)]TJ/F27 11.955 Tf 11.956 0 Td[(ti.Proof.WerstnotethatLnft=nXi=1pntfti t)]TJ/F27 11.955 Tf 11.956 0 Td[(tip0nti:.1.10Nextwenotethatthefunctionfz z)]TJ/F27 11.955 Tf 11.955 0 Td[(tpnzhasresidueft pntatz=t,andresiduefti ti)]TJ/F27 11.955 Tf 11.955 0 Td[(tp0ntiatz=ti.Bytheresiduetheoremitthenfollowsthat1 2iZ)]TJ/F27 11.955 Tf 28.164 18.881 Td[(fz z)]TJ/F27 11.955 Tf 11.955 0 Td[(tpnzdz=ft pnt+nXi=1fti ti)]TJ/F27 11.955 Tf 11.955 0 Td[(tp0nti:Multiplyingthroughbypntweobtain1 2iZ)]TJ/F27 11.955 Tf 15.743 18.881 Td[(fzpnt z)]TJ/F27 11.955 Tf 11.955 0 Td[(tpnzdz=ft+nXi=1ftipnt ti)]TJ/F27 11.955 Tf 11.955 0 Td[(tp0nti:.1.11Inspectionof.1.10and.1.11thengivesusourdesiredresult. Thereisanotherestimateforinterpolationerrorwhichwewillneedinourhierarchysection.WeneedtodiscussNewtonInterpolationbeforepresentingthisestimateinterpolationonLejasequencesisofcourseanexampleofNewtoninterpolation,butwedonotneedthisestimateforourresultonLejasequences.Newtoninterpolation,werecall,iswhenwearegivenasequencefxngandforeachn,Lnfagreeswithfontherstntermsofthissequence.Inthisscenario,ifpnisthepolynomialobtainedbyinterpolatingontherstn+1terms,andpn)]TJ/F25 7.97 Tf 6.586 0 Td[(1isthepolynomialobtainedbyinterpolatingontherstnterms,wemusthavepnx=pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1x+an+1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x2x)]TJ/F27 11.955 Tf 11.955 0 Td[(xn:.1.12Indeed,sincepn)]TJ/F25 7.97 Tf 6.586 0 Td[(1agreeswithfatx1;x2;:::;xn,therighthandsideof.1.12alsomustagreewithfatx1;x2;:::;xn.Thereforebyasuitablechoiceofan+1,therighthandsideof.1.12willalsoagreewithfatxn+1.ItfollowsthatforNewtonInterpolation,wehavethefollowingrepresentationoftheinterpolationpolynomialLn+1f=a1+a2x)]TJ/F27 11.955 Tf 11.955 0 Td[(x1+a3x)]TJ/F27 11.955 Tf 11.955 0 Td[(x1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x2+.1.13+an+1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x2x)]TJ/F27 11.955 Tf 11.956 0 Td[(xn:6

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Theconstantsaiin.1.13arecalleddivideddierences,andthefollowingnotationisoftenusedan=[x1;x2;:::;xn]f;n=1;2;:::;Itisobviousthata1=[x1]f=fx1,andaneasycalculationgivesthata2=[x1;x2]f=fx1)]TJ/F27 11.955 Tf 11.956 0 Td[(fx2 x1)]TJ/F27 11.955 Tf 11.956 0 Td[(x2:Ingeneralwehavethefollowingusefulandeasilyprovenformulasee[5]page98[x1;x2;:::;xn]f=[x2;x3;:::;xn]f)]TJ/F19 11.955 Tf 11.955 0 Td[([x1;x2;:::;xn)]TJ/F25 7.97 Tf 6.586 0 Td[(1]f xn)]TJ/F27 11.955 Tf 11.955 0 Td[(x1:.1.14Infact,.1.14iswherethetermdivideddierencecomesfrom.Thus,intermsofdivideddierences,NewtoninterpolationtakestheformLn+1f=[x1]f+[x1;x2]fx)]TJ/F27 11.955 Tf 11.955 0 Td[(x1+[x1;x2;x3]fx)]TJ/F27 11.955 Tf 11.955 0 Td[(x1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x2++[x1;x2;:::;xn+1]fx)]TJ/F27 11.955 Tf 11.955 0 Td[(x1x)]TJ/F27 11.955 Tf 11.955 0 Td[(x2x)]TJ/F27 11.955 Tf 11.955 0 Td[(xn:.1.15Wenowillustratehowtocalculatedivideddierenceswithanexamplewhichwillbeofparticularinteresttouslateron.Wecalculatethedivideddierencesoffz=1 z)]TJ/F27 11.955 Tf 11.955 0 Td[(:Wehave[x1]f=1 x1)]TJ/F27 11.955 Tf 11.955 0 Td[(and;[x1;x2]f=[x2]f)]TJ/F19 11.955 Tf 11.956 0 Td[([x1]f x2)]TJ/F27 11.955 Tf 11.956 0 Td[(x1=1 x2)]TJ/F28 7.97 Tf 6.586 0 Td[()]TJ/F25 7.97 Tf 21.505 4.707 Td[(1 x1)]TJ/F28 7.97 Tf 6.586 0 Td[( x2)]TJ/F27 11.955 Tf 11.955 0 Td[(x1=)]TJ/F19 11.955 Tf 9.298 0 Td[(1 x1)]TJ/F27 11.955 Tf 11.956 0 Td[(x2)]TJ/F27 11.955 Tf 11.955 0 Td[(:Wenowuse[x1;x2]fand[x1;x3]ftocalculate[x1;x2;x3]f.Wehavethefollowing[x1;x2;x3]f=[x1;x2]f)]TJ/F19 11.955 Tf 11.955 0 Td[([x1;x3]f x2)]TJ/F27 11.955 Tf 11.955 0 Td[(x3=)]TJ/F25 7.97 Tf 6.587 0 Td[(1 x1)]TJ/F28 7.97 Tf 6.586 0 Td[(x2)]TJ/F28 7.97 Tf 6.586 0 Td[()]TJ/F30 7.97 Tf 35.27 4.707 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1 x1)]TJ/F28 7.97 Tf 6.587 0 Td[(x3)]TJ/F28 7.97 Tf 6.587 0 Td[( x2)]TJ/F27 11.955 Tf 11.955 0 Td[(x3=1 x1)]TJ/F27 11.955 Tf 11.955 0 Td[(x2)]TJ/F27 11.955 Tf 11.955 0 Td[(x3)]TJ/F27 11.955 Tf 11.956 0 Td[(:Itisclearthatingeneralwewillhave[x1;x2;:::;xn]f=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n+1 x1)]TJ/F27 11.955 Tf 11.955 0 Td[(x2)]TJ/F27 11.955 Tf 11.955 0 Td[(xn)]TJ/F27 11.955 Tf 11.955 0 Td[(:Sincethiscalculationwillbeimportantlaterinthedissertation,werecorditasaLemma.7

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Lemma2.1.2Letfxngbeasequenceofdistinctcomplexnumbersandletf=1 z)]TJ/F27 11.955 Tf 11.955 0 Td[(;whereisdierentfromeveryxn.Thenthen-thdivideddierenceoffisgivenby[x1;x2;:::;xn]f=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n+1 x1)]TJ/F27 11.955 Tf 11.955 0 Td[(x2)]TJ/F27 11.955 Tf 11.955 0 Td[(xn)]TJ/F27 11.955 Tf 11.955 0 Td[(:HavinghadthisdiscussionofNewtoninterpolationanddivideddierences,wearenowreadytogivethesecondinterpolationerrorformulawhichwewillneed.Thisisaclassicalresultseee.g.[5],page100,butwepresentaproof.Lemma2.1.3Letfxngbeasequenceofdistinctpointsontheplane,andletxbeapointdierentfromeveryxi.Thenwehavefx)]TJ/F27 11.955 Tf 11.955 0 Td[(Ln+1f;x=[x1;x2;:::;xn+1;x]fn+1Yi=1x)]TJ/F27 11.955 Tf 11.955 0 Td[(xi:Proof.Lettbeanarbitrarynodenotequaltoanyofthex1;x2;:::;xn+1.Thenwehavepn+1f;x1;x2;:::;xn+1;t;x=pnf;x1;:::;xn+1;x++[x1;x2;:::;xn+1;t]fn+1Yi=1x)]TJ/F27 11.955 Tf 11.955 0 Td[(xi:Nowputx=t.Sincethepolynomialontheleftinterpolatestofatt,wegetft=pnf;t+[x1;x2;:::;xn+1;t]fn+1Yi=1t)]TJ/F27 11.955 Tf 11.955 0 Td[(xi:Writingagainxfortwhichwasarbitrary,afterall,wendfx)]TJ/F27 11.955 Tf 11.955 0 Td[(pnf;x=[x1;x2;:::;xn+1;x]fn+1Yi=1x)]TJ/F27 11.955 Tf 11.955 0 Td[(xi: 2.2PotentialTheoryThefocusofthisdissertationisonLagrangeinterpolationratherthanpotentialthe-ory.However,thereisaconnectionbetweenLagrangeinterpolationandpotentialtheorywhichwewillneedtoexploittoobtainourresults.Whatfollowsarethebasicdenitionsandclassicalresultsfrompotentialtheorywhichareofinteresttousinthisdissertation.Asthefocusofthisdissertationisnotpotentialtheory,theresultsaregivenwithoutproof.Astandardreferenceforpotentialtheoryintheplanearethebooks[9]and[12].8

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Denition2.2.1LetbeaniteBorelmeasureonCwithcompactsupport.ItspotentialisthefunctionUz:C![;1;denedbyUz=Zlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(wjdwz2C:Inthisdissertationwewilloftenhavesequencesofmeasureswhichareconvergentintheweakstarsense,andwewillneedtomakeuseofthefollowingTheorem.Theorem2.2.2[9],page59LetKbeacompactsetandletnbeasequenceofmeasureswhichconvergeweakstartosomemeasure.Thenthefollowingistruelimsupn!1UnzUz;z2C:Wewillalsoneeduppersemi-continuouspropertiesofpotentials,givenbythefollowingdenitionandTheorem.Denition2.2.3Afunctionf:C![;1iscalleduppersemi-continuousifforallthesetU=fz:fz
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Denition2.2.7LetKbeacompactsubsetofC,anddenotebyMKthecollectionofallBorelprobabilitymeasuresonK.IfthereexistsK2MKsuchthatIK=sup2MKI;thenKiscalledanequilibriummeasureforK.Theorem2.2.8[9],page58Everynon-polarcompactsetKhasauniqueequi-libriummeasure.Belowweshallintroducethenotionoflogarithmiccapacity,andwithitasetisnon-polarifandonlyifitisofpositivecapacity.Asstatedabove,equilibriumdistributionswillbeofparticularinteresttous.Wewillneedthefollowingimportanttheoremregardingequilibriumdistributions.Theorem2.2.9[9],page59LetKbeacompactsetinC,andletKbetheequilibriummeasureforK.ThenUKzIK;z2C:SinceUKisharmoniconCnK,andthusbytheminimumprincipleforharmonicfunctionsdoesnotattainaminimumonCnK,asadirectconsequenceofTheorem2.2.9,wehaveTheorem2.2.10LetUKbetheequilibriumdistributionforK.ThenUKz>IK;z2CnK:Wenowintroduceourdenitionofregularity.Ourdenitionisnotoneofthestandarddenitionsgiveninliterature.Wearehoweverusingthisdenitionbecauseitisequivalenttothestandardoneusedintheliterature,andbecauseitsimpliesourdiscussion.Denition2.2.11AcompactsetKiscalledregularifUKz=IK;z2K:Typically,regularityisdenedintermsoftheDirichletproblem:namelyitisrequiredthattheDirichletsolutiontoacontinuousboundarydatabecontinuousontheclosureofthedomain.Ourdenitionisequivalenttothatone,butitismoreconvenienttouse.Asexamplesofregularsetsconsidercompactsetsofatleasttwopointsforwhichthecomplementissimplyconnected.WealsohavethefollowingTheoremregardingregulardomains.Theorem2.2.12[12],page54LetKbearegulardomainwithequilibriumdis-tributionK.ThenUKzisacontinuousfunctiononK,andhenceonthewholecomplexplane.10

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Wegiveonelastdenitioninthissection.Theconceptofcapacitywillbeusedthroughoutthisdissertation.Denition2.2.13Thelogarithmiccapacityofnon-polarcompactsetKisgivenbyCapK=expIK:Throughoutthisdissertationwewillbeconvertingpolynomialstopotentialsbytakingthenaturallogofthesepolynomials,andthenconvertingtheselogsintointe-grals.Wewilldiscussthismoreinoursubsectiononequilibriummeasures,butthereadershouldmakenoteofDenition2.2.11andDenition2.2.13.Indeed,manyofourintegralswillbesetequaltologCapK.2.3LejaPointsWerecallfromtheintroductionthatLejasequencesassociatedwithacompactsubsetKoftheplanearedenedinductively.Letz12Kbearbitrary.Oncez1;z2;:::;zn)]TJ/F25 7.97 Tf 6.587 0 Td[(1havebeendetermined,znischosensothatVz=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Yi=1jz)]TJ/F27 11.955 Tf 11.955 0 Td[(zijismaximizedforz=zn.WespeakofaLejasequence,ratherthantheLejasequence,sincetheremaybemorethanonechoiceforzn.Infact,thechoiceofz1isarbitrary.SinceLejapointsonacompactsetKaredeterminedbyndingthemaximumofapolynomialonK,bythemaximummodulustheorem,itfollowsthataLejasequenceonacompactsetKwillnecessarilylieonK'souterboundary,wheretheouterboundaryofKisdenedastheboundaryoftheunboundedcomponentofthecomplementCnK.Inourhierarchyofinterpolationschemes,oneoftheassumptionsthatwemakeonthesystemofnodesisthattheylieontheouterboundaryofK,whichisthenautomaticallysatisedforLejapoints.InprovingthattheLebesgueconstantsassociatedwithLejasequencesonacloseddomainaresubexponential,ourproblemwillbereducedtoprovingthattheLebesgueconstantsassociatedwithLejasequencesonarcsaresubexponential.AnothereasilyseenpropertyofLejasequences,andonewhichweshalluse,isthattheirselectionisinvariantwithrespecttorotationsandtranslationsintheplane.Ifz1;:::;znaretherstnpointsofaLejasequenceonacompactsetK,andifthecomplexplaneisrotatedaboutapointwthroughanangleof,thenanewLejasequencewillbeobtainedfromthenewKsimplybyrotatingtheoriginalLejapointsaboutthepointwthroughanangleof.Similarly,ifKistranslatedbytheconstantc,thenanewLejasequencewillbez1+c;z2+c;:::.ThemostimportantpropertyofLejapointswhichweshallneedisnoteasilyseen.ThefollowingTheoremwillbeneededinourmainresult.ItregardstheconnectionbetweenLejasequencesandequilibriumdistributions.Sincethefocusofthisdisserta-tionisLagrangeinterpolation,andwearesimplyusingpotentialtheory,thisTheoremisgivenwithoutproof.It'sproofcanbefoundin[12],page258].11

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Theorem2.3.1LetKbeacompactsetwithequilibriumdistributionK.LetfzjgbeaLejasequenceonK,andletnbethenormalizedcountingmeasurewithrespecttotherstntermsofthisLejasequence,i.e.,n=1 nnXj=1zj:Thenn*K:2.4EquilibriumMeasuresAsstatedinsubsection1.2,thereisaconnectionbetweenpotentialtheoryandpolyno-mialinterpolation.HavingintroducedLejapointsaswellasoneofthemajortheoremswhichwewillexploitTheorem2.3.1,wearenowreadytoexploretheconnectionbetweenpotentialtheoryandpolynomials.InTheorem2.3.1welinkedLejasequencestoequilibriummeasuresthroughtheconceptofweakstarconvergence.Hereagain,wewilldiscussweakstarconvergence,asitwillfacilitatemuchofourinterplaybetweenpolynomialsandpotentialtheory.Toseethishowthiswillbedoneconsiderthattheabsolutevalueofapolynomialpzwithzerosatz1;z2;:::;zn2Kcanbewrittenasjpzj=nYj=1jz)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj=expn1=nnXj=1logjz)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj!.4.16=expnZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt;wherenistheprobabilitymeasuretakingthevalue1=natt=z1;:::;zn.Ontheleftsideof2.4.16wehaveapolynomial,whileontherighthandsidewehavetheexponentialfunctionraisedtoapotential,Unz.Thusndingwhereapolynomialiseitherlargeorsmallisequivalenttondingwhereapotentialiseitherlargeorsmall.Further,whentheprobabilitymeasuresnassociatedwithoursequenceofpointsfzjgconvergeweakstartoameasure,ndingwhereapolynomialiseitherlargeorsmallwillbeequivalenttondingwherethepotentialofislargeorsmall.Inmanycasesourmeasures,n,willconvergeweakstartotheequilibriummeasureKonKasofcourseisthecasewhenweareconsideringLejasequences.Inthissubsectionwegiveseveralresultsregardingequilibriummeasureswhichwillhelpusinourdiscussionofpolynomialinterpolation.Whileweomittedproofsinsection1:2,herewewillgiveproofs.Therearetworeasonsforthis.Therstreasonisthatmanyoftheseresultsaretoosimpletobefoundintheliteratureonpotentialtheory.Thesecondreasonisthatwewanttoillustratetheimportanceof2.4.16,therelationshipbetweenpolynomialsandpotentials.Beforebeginningwegiveawordaboutnotation.Inwhatfollows,ifwearegivenasequenceofpointsfzjg,thennwilldenotetheprobabilitymeasurewhichtakesthe12

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value1=nattherstnpoints,i.e.n=1=nnXj=1zj:Asbefore,wewilluseKtodenotetheequilibriummeasureonacompactsetK.WealsonotethatthereadershouldrememberthedenitionofapotentialassociatedwithagivenmeasureonacompactsetK,Uz=ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdt:.4.17ThisisbecausewewilloftennotusetheexplicitnotationUzinthefollowinglemmas,butonlytherighthandsideof.4.17.Lemma2.4.1LetKbeacompactsetandletbeaprobabilitymeasureonKsuchthat6=K.LetM=z2K:ZKlogjz)]TJ/F27 11.955 Tf 11.956 0 Td[(tjdt0,andfurthermore,thereexistsadiscDwithcenterinMsuchthatD>0andsuchthatforsome>0thefollowingholdsZKlogjz)]TJ/F27 11.955 Tf 11.956 0 Td[(tjdt0thefollowingholdsZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdt0,wemusthaveD>0foroneoftheD'scontainedinthecountablesubcover.Thiscompletestheproof. Lemma2.4.2LetKbearegularcompactset,andletbeaprobabilitymeasureonKsuchthat6=K.Thenthereexistsz2KsuchthatZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdtlogCapK:13

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Proof.SinceKisregular,wehavethefollowingZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt=logCapK;z2K;andthusthatZKZKlogjz)]TJ/F27 11.955 Tf 11.956 0 Td[(tjdKtdz=logCapK:ByanapplicationofFubini'sTheorem,wemustalsohaveZKZKlogjz)]TJ/F27 11.955 Tf 11.956 0 Td[(tjdtdKz=logCapK:.4.18NowifZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdt0suchthatthefollowingholdsZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt>logCapK+;z2)]TJ/F27 11.955 Tf 7.314 0 Td[(:Proof.SinceKisregular,byTheorem2.2.12UKz=ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKtiscontinuouson.Further,UKz>logCapK;z2)]TJ/F27 11.955 Tf 7.314 0 Td[(:seeTheorem2:2:10Thusforeachz2)-326(thereexistsaneighborhoodNzandanNzsuchthatUKw>logCapK+Nz;w2Nz:Since)-366(iscompactweneedonlynitelymanyNz'stocover.ThesmallestoftheNz'sisourdesiredepsilon. 14

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Lemma2.4.4LetKbearegularcompactsetwithequilibriumdistributionK,letfngbeasequenceofprobabilitymeasureswhichconvergeweakstartoK,andlet)]TJ/F44 11.955 Tf -427.926 -13.948 Td[(beaniteunionofJordancurvescontainingK.Thenthereexists>0,andanNsuchthatforn>NwehavethefollowingZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt>logCapK+;z2)]TJ/F27 11.955 Tf 7.314 0 Td[(:Proof.Let1beasinLemma2.4.3.Thensincelogjz)]TJ/F27 11.955 Tf 12.229 0 Td[(tjiscontinuousonKforxedz2,wehavethefollowingZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt!ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt>logCapK+1:.4.20Further,sincelogjz)]TJ/F27 11.955 Tf 11.188 0 Td[(tjiscontinuouson)]TJ/F20 11.955 Tf 94.699 0 Td[(K,ifwexz2)-295(and2>0,thenthereexistsasmallneighborhoodDzofzsuchthatforallprobabilitymeasuresmwehaveZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdmt)]TJ/F32 11.955 Tf 11.955 16.272 Td[(ZKlogjw)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdmt<2;w2Dz:.4.21By2.4.20,foreachz2)-326(wecanndanNzsuchthatforn>NzwehaveZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt>logCapK+1=2:.4.22Andnowby.4.21and2.4.22taking2=1=4,thereexistsDzsuchthatZKlogjw)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt>logCapK+1=4;n>Nz;w2Dz:Since)-342(iscompact,weneedonlynitelymanyDz'stocover.TakingNtobethelargestoftheNz'sandtobe1=4,wehavecompletedourproof. Lemma2.4.5LetKbeacompactset,KtheequilibriumdistributionforK,andletfngbeasequenceofmeasureswhichconvergeweakstartoK.Thenforall>0thereexistsandanN,suchthatnV<;forn>N;wheneverdiameterV<:Proof.Let>0begiven.Thenthereexists1suchthatdiameterV<1impliesKV< 2:Toseethis,considerthatanimmediateconsequenceofthedenitionofKtellsusthatKfzg=0;forallz:.4.2315

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From2.4.23itfollowseasilybecauseKhasniteenergyandhencenopointmassesthateachpointzhasaneighborhoodNzsuchthatKNz<=2:BycompactnessofKthereexistNz1;:::;NzmwhichcoverK.Associatedwiththiscoveristhena1suchthatanydiscofradius1andwithcenterinKiscontainedinoneoftheNzi.Thisisourrequired1.CoverKwithdiscsD1;:::;Dmofradii1=2andcentersatsomexi.Thenthereexistcontinuousfunctionsfisuchthatfix=1ifx2Di0ifjx)]TJ/F27 11.955 Tf 11.955 0 Td[(xij>1Sincen*k,thereexistsNsuchthatn>NimpliesthatZKfidnNimpliesthatnDi<.SinceSDicoversK,thereexistsasuchthatallsetsofdiameterarecontainedinsomeDi.Thisisourrequireddelta. InasimilarlineofthoughttoLemma2.4.5,wealsohavethefollowingLemma.Lemma2.4.6LetKbearegularcompactsetwithequilibriumdistributionK.Thenforall>0,thereexistssuchthatZjt)]TJ/F28 7.97 Tf 6.587 0 Td[(zj
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Lemma2.4.7LetKbeacompactsetsuchthatKistheunionofnitelymanyJordanarcs,letKbetheequilibriumdistributiononK,andletfngbeasequenceofmeasuresonKwhichconvergeweakstartoK.Thenn\051!K\051forallsubarcs)]TJ/F20 11.955 Tf 10.635 0 Td[(K.Proof.Let>0begiven.Wecancovertheendpointsof)-369(inopendiscsD1;D2suchthatKD1+KD2< 2;.4.24andsuchthatforsomeN1,wehavenD1+nD2< 2;n>N1:.4.25Wenotethat.4.25isduetoLemma2.4.5.ByUrysohn'sLemma,thereexistsacontinuousfunctionfofnorm1suchthatf=1on)-418(andsuchthatf=0onKn)]TJ/F32 11.955 Tf 13.859 8.967 Td[(SD1SD2.SincenconvergesweakstartoK,thereexistsN2>N1suchthatn>N2impliesZKftdnt)]TJ/F32 11.955 Tf 11.955 16.272 Td[(ZKftdt< 2:.4.26Comparisonof.4.24,.4.25,and.4.26,andthedenitionofftellsusthatforn>N2wehavejn\051)]TJ/F27 11.955 Tf 11.955 0 Td[(\051j<:Thiscompletestheproof. AsadirectconsequenceofLemma2.4.7,wehavethefollowingLemma.Lemma2.4.8LetKbeasinLemma2:4:7.Letsbeasimplestep-functiondenedonK,takingitsnitelymanyvaluesa1;:::;ajonthesubarcsA1;:::;Aj.Also,letfngbeasequenceofmeasuresonKwhichconvergeweakstartoK.ThenwehaveZKstdnt!ZKstdt:AnotherdirectconsequenceofLemma2.4.7isthefollowinglemma.Thislemmawillbeimportantinourmainresult.Lemma2.4.9LetK=[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1],letfxjgbeasequenceofLejapointson[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1],letKbetheequilibriumdistributionon[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1],andletnbetheprobabilitymeasuretakingthevalue1=nattherstnLejapoints.NowletR1begivenandletx2;1besuchthat1)]TJ/F19 11.955 Tf 12.394 0 Td[(2R)]TJ/F27 11.955 Tf 12.394 0 Td[(x>0.Finally,letkbethenumberoftherstnLejapointscontainedintheintervalI=[1)]TJ/F19 11.955 Tf 12.13 0 Td[(2R)]TJ/F27 11.955 Tf 12.13 0 Td[(x;1)]TJ/F19 11.955 Tf 12.13 0 Td[(21)]TJ/F27 11.955 Tf 12.13 0 Td[(x]:ThenthereexistsNsuchthatforn>Nwehavek=n>1 4Z1)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F28 7.97 Tf 6.587 0 Td[(x1)]TJ/F25 7.97 Tf 6.586 0 Td[(2R)]TJ/F28 7.97 Tf 6.587 0 Td[(x1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2dt:17

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Proof.Theequilibriumdistributionfor[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1]is[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]=1 21 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2:Sincen*[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]seeTheorem2.3.1,byLemma2.4.7wehavek=n=nI![)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]I=1 2Z1)]TJ/F25 7.97 Tf 6.586 0 Td[(21)]TJ/F28 7.97 Tf 6.586 0 Td[(x1)]TJ/F25 7.97 Tf 6.587 0 Td[(2R)]TJ/F28 7.97 Tf 6.586 0 Td[(x1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2dt:ThusthereexistsNsuchthatn>Nimpliesk=n>1 4Z1)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F28 7.97 Tf 6.587 0 Td[(x1)]TJ/F25 7.97 Tf 6.586 0 Td[(2R)]TJ/F28 7.97 Tf 6.587 0 Td[(x1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2dt: Whenwehaveasequenceofmeasuresfngwhichconvergeweakstartoameasure6=K,thenLemma2.4.7doesnotapply,butwedohavethefollowingLemma,whichwewillneedinourHierarchysection.Lemma2.4.10LetKbeacompactset,letbeameasureonK,andletfngbeasequenceofmeasureswhichconvergeweakstarto.NowletDbeanopendiscsuchthatD>0.ThenthereexistsNsuchthatn>NimpliesnD>0:Proof.LetRbetheradiusofDandassumew.l.o.g.thatDiscenteredattheorigin.Denefz=1)-175(jzj=RifjzjRandfz=0otherwise.Thenfiscontinuous,andbyassumptionZfd>0:Hence,byweakstarconvergence,forlargenZfdn>0;whichprovestheclaim. Wewillnowpresenttwotheoremswhichwewilluseinourmainresult.Oneofthesewillalsobeimportanttousinourhierarchysection.Theproofsofeachofthesetheoremsillustratetheimportanceofpotentialtheorytothisdissertation.Ineachproofweeitherconvertapolynomialoraproductintoapotential,andthenusepotentialtheorytoachieveourresults.Theorem2.4.11LetKbearegularcompactset,letfzjgbeasequenceofLejapointsonK,andletPn;kz=Qnj=1;j6=kz)]TJ/F27 11.955 Tf 11.846 0 Td[(zj.ThenkPn;kk1=nK!CapKuniformlyink.18

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Proof.WerstshowthatlimsupnkPn;kk1=nKCapK:.4.27Wenotehowever,thatshowing2.4.27isequivalenttoshowinglimsupnkPn;kk1=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1KCapK;.4.28andthattoshow.4.28,itisenoughtoverifythatforalln;kthereexistsz2KsuchthatlogjPn;kzj1=n)]TJ/F25 7.97 Tf 6.587 0 Td[(1logCapK:.4.29Toshow.4.29wewritelogjPn;kzj1=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1=ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdn;kt;.4.30wheren;kisthemeasurewhichtakesthevalue1 n)]TJ/F25 7.97 Tf 6.586 0 Td[(1atthepointst=zj;j6=k,1jn.Sincen;kisnottheequilibriumdistributionforK,byLemma2.4.2theintegralin.4.30isindeedgreaterthanorequaltologCapKforsomez2K.Thisproves.4.27.Havingshown.4.27,tocompletetheproofitsucestoshowthatforallthereexistsN,chosenindependentlyofk,suchthatn>NimpliesthatkPn;kk1=nK0thereexistsaniteunionofJordancurves)-298(containingK,andanNsuchthatforn>NkPn;kk1=n)]TJ/F27 11.955 Tf 17.426 3.241 Td[(0,thereexistsa)-326(containingKandanNsuchthat1 nlogjPn;kzjN;z2)]TJ/F27 11.955 Tf 7.314 0 Td[(:.4.33Thisdoesindeedshow.4.32.Toseethisreplacethein.4.33withlog+,whereissuchthatCapK<.Toshow2.4.33werstwrite1 nlogjPn;kzj=1 nnXj6=klogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj=n)]TJ/F19 11.955 Tf 11.956 0 Td[(1 nZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdn;kt:.4.34Next,sincen*K;19

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thisisbyTheorem2.3.1whereKistheequilibriumdistributionforK,andnistheprobabilitymeasuretakingthevalue1=natt=zj;j=1;2;:::n;itisclearthatn;k*K:.4.35Further,sinceKisregular,byTheorem2.2.12UKz=ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKtiscontinuousinC,andbydenitionofregularityUKz=logCapKforz2K.Itfollowsthatwecanchoose)-326(sucientlyclosetoKsuchthatZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKtNimplies1 nlogjPn;kzjmaxfN1;:::;Nmg;wewillhave1 nlogjPn;kzjN;z2)]TJ/F27 11.955 Tf 7.314 0 Td[(:Wehavethusshown.4.33andcompletedourproof. Theorem2.4.12LetKbetheunionofnitelymanyJordanarcs,letfzjgbeasequenceofLejapointsonK,andfor1knsetPn;k;=Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkj;1jnjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj:Thenforall>0thereexists,N,suchthatforn>NwehavethefollowingPn;k;1=n)]TJ/F19 11.955 Tf 11.955 0 Td[(CapK<;k=1;2;:::;n:20

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Proof.Ratherthandirectlyprovingourdesiredresult,wewillinsteadshowthatforall>0thereexists,N,suchthatforn>NwehavelogPn;k;1=n)]TJ/F19 11.955 Tf 11.956 0 Td[(logCapK<;k=1;2;:::;n:Thetwostatementsclearlyimplyeachother.Wewrite1 nlogPn;k;=1 nXjzj)]TJ/F28 7.97 Tf 6.587 0 Td[(zkjlogjzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zkj=Zjzk)]TJ/F28 7.97 Tf 6.586 0 Td[(tjlogjt)]TJ/F27 11.955 Tf 11.955 0 Td[(zkjdnt;wherenistheprobabilitymeasureconcentratedontherstnLejapoints,givingtheweight1=ntoeachofthesepoints.OurassumptionsaboutKimplythatKisregular,andwethushaveZKlogjzk)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt=logCapK;whereKistheequilibriumdistributionforKthisisbyourdenitionofregularity.ByLemma2.4.6wealsoknowthatZjzk)]TJ/F28 7.97 Tf 6.586 0 Td[(tjlogjzk)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt!ZKlogjzk)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt=logCapK;uniformlyinkas!0.ItfollowsthatwewillhavecompletedourproofifwecanshowthatZjzk)]TJ/F28 7.97 Tf 6.587 0 Td[(tjlogjt)]TJ/F27 11.955 Tf 11.955 0 Td[(zkjdnt!Zjzk)]TJ/F28 7.97 Tf 6.587 0 Td[(tjlogjzk)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt;andthatthisconvergenceoccursuniformlyink.Withthisinmind,wedeneforx;t2Kgxt=logjx)]TJ/F27 11.955 Tf 11.956 0 Td[(tjifjx)]TJ/F27 11.955 Tf 11.956 0 Td[(tj0elsewhereThenbyourassumptionsonK,forallx2K,thereexistsimplefunctions sx;sx ,whichtaketheirnon-zerovaluesonarcscontainedinK,andthereexistsadiscDx>0centeredatxsuchthatthefollowingistrue:1:sx tgyt sxt;y2Dx;t2K2:RK sxtdKt)]TJ/F27 11.955 Tf 11.955 0 Td[(RKgytdKsx tdKt+;y2DxSinceLemma2.4.8tellsusthatZK sxtdnt!ZK sxtdKt;andthatZKsx tdnt!ZKsx tdKt;21

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itfollowsthatthereexistsNxsuchthatn>NximpliesthatZKgytdnt)]TJ/F32 11.955 Tf 11.955 16.272 Td[(ZKgytdKt<3;y2DxNowsinceKiscompact,itfollowsthatthereexistsx1;x2;:::;xm2KsuchthatKSDxi.ThenforN=maxfNx1;Nx2;:::;Nxmg,andn>NwehaveZKgytdnt)]TJ/F32 11.955 Tf 11.955 16.273 Td[(ZKgytdKt<3;y2K:Basedonourdenitionofgyt,itfollowsthatwehaveshownthatZjxk)]TJ/F28 7.97 Tf 6.587 0 Td[(tjlogjt)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjdnt!Zjxk)]TJ/F28 7.97 Tf 6.587 0 Td[(tjlogjxk)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKtuniformlyink:Thiscompletesourproof. 22

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3HierarchyofInterpolationSchemesWithourpreliminariescompleted,wearereadytogiveourrstresult,ahierarchyofinterpolationschemes.LetKbearegularcompactsetandletfzn;kg1kn;n=1;2;:::beatriangulararrayofpointslyinginK.LetLnfbetheLagrangeinterpolationtofbasedonthepointsinthen-throw:Lnfz=Lnf;z=nXk=1fzn;kln;kz:Weshallalsousethenormalizedcountingmeasuresonthepointsinthen-throw:n=1 nnXk=1zn;k:Werecallthenotionofouterboundary.ThecomplementCnKconsistsofcon-nectedcomponents,oneofthem,denotedby,isunbounded.Nowtheboundary@ofiscalledtheouterboundaryofK.Weshallassumeofthepointszn;kthattheylieontheouterboundaryofK.Considerthefollowingstatements:iTheLebesgueconstantsnassociatedwithLnaresubexponential:1=nn!1;iin!Kintheweakstarsense,whereKistheequilibriumdistributionofK,iiiLnf!funiformlyonKforallfunctionsfwhichareanalyticonandinsideaniteunionofclosedcontours)]TJ/F28 7.97 Tf 164.825 -1.793 Td[(i;i=1;2:::;v,suchthatK)-278(=[)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(i:,ivLnf!funiformlyonKforallfoftheformfz=1=z)]TJ/F27 11.955 Tf 11.956 0 Td[(,2.Clearly,iiiimpliesiv,butactuallywenowshowthatii-ivareequivalent,andiimpliesanyofthem.Theorem3.0.13LetKbeacompactsetwithpositivecapacity,andassumethatallzn;klieontheouterboundaryofK.Thenii,iii,andivareequivalent,andiimpliesanyofthem.23

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3.1ProofofiiiWeshowthatifndoesnotconvergeweakstartoKthentheLebesgueconstantsassociatedwithfzn;kgarenotsubexponential.IffngdoesnotconvergeweakstartoK,thenbyHelly'sselectiontheorem,wecanndasubsequencefnsgsuchthatns*,whereissomeprobabilitymeasurenotequaltoK.Letusrelabelthissubsequenceasfng.Recallingthatn=supz2KnXj=1Qnj=1jz)]TJ/F27 11.955 Tf 11.955 0 Td[(zn;jj Qnj=1jzn;k)]TJ/F27 11.955 Tf 11.956 0 Td[(zn;jj!;.1.1theideaofourproofwillbetouseresultsfrompotentialtheorytosomewhateasilycompleteourproof.Specically,wewillndasequencefzngK,andasequencefzn;kg,sothatnYj=1;j6=kjzn)]TJ/F27 11.955 Tf 11.956 0 Td[(zn;jj!1=n=exp1 nnXj6=klogjzn)]TJ/F27 11.955 Tf 11.955 0 Td[(zn;jj!=expZKlogjzn)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt;islargeandsothatnYj=1;j6=kjzn;k)]TJ/F27 11.955 Tf 11.956 0 Td[(zn;jj!1=n=exp1 nnXj6=klogjzn;k)]TJ/F27 11.955 Tf 11.955 0 Td[(zn;jj!=expZKlogjzn;k)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdnt;issmall.Thequotientofthesetwotermswillthentellusthat.1.1isnotsubex-ponential.Werstndoursequencefzn;kgandwebegintodothisbydeningthesetM=z2K:ZKlogjz)]TJ/F27 11.955 Tf 11.956 0 Td[(tjdt0andsuchthatZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdt0.SinceD>0;byLemma2.4.10thereexistsNsuchthatforn>NwehavenD>0:24

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Inparticular,forn>Nwecanalwaysndazn;k2D.Foreachnwechoosesuchazn;kandweformthemeasuresn=1 n)]TJ/F19 11.955 Tf 11.955 0 Td[(1Xj=1;j6=kznn;j:Sincenconvergesweakstarto,itisclearthatnmustconvergeweakstarto.Nowbyourchoiceofzn;k,wehaveZKlogjzn;k)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdt1:Thiscompletestheproof. 25

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3.2ProofofiiiiiLet)-337(beaunionofnitelymanysmoothJordancurveswhichcontainKandwhicharecontainedthemselvesinf'sdomainofanalyticity.ThenbytheremaindertheoremTheorem2.1.1wehaveforz2Kfz)]TJ/F27 11.955 Tf 11.955 0 Td[(Lnfz=fz)]TJ/F28 7.97 Tf 18.02 14.944 Td[(nXj=1fzn;jPnz P0nzjz)]TJ/F27 11.955 Tf 11.955 0 Td[(zn;j=1 2iZ)]TJ/F27 11.955 Tf 14.871 18.881 Td[(ftPnz t)]TJ/F27 11.955 Tf 11.955 0 Td[(zPntdt;wherePnz=Qnj=1z)]TJ/F27 11.955 Tf 11.055 0 Td[(zn;j.Thus,inordertoshowthatLnfconvergesuniformlytof,itsucestoshowthatZ)]TJ/F27 11.955 Tf 14.871 18.881 Td[(ftPnz t)]TJ/F27 11.955 Tf 11.955 0 Td[(zPntdt!0.2.5uniformlyinz2K.Nowtoshow3.2.5,sinceweknowthatftisboundedon,andthatt)]TJ/F27 11.955 Tf 11.725 0 Td[(zisboundedon)]TJ/F20 11.955 Tf 72.128 0 Td[(K,wewillfocusonthequotientPnz=Pnt.Usingpotentialtheory,wewillndanupperboundforjPnzj1=n;z2K;andalowerboundforjPntj1=n;t2)]TJ/F27 11.955 Tf 7.314 0 Td[(;andwewillthencompleteourproofbyshowingthatPnz Pnt=jPnzj1=n jPntj1=nn0andN1suchthatforn>N1wehaveZKlogjt)]TJ/F27 11.955 Tf 11.955 0 Td[(zjdnz>logCapK+;t2)]TJ/F27 11.955 Tf 7.314 0 Td[(:WethenhavethatjPntj1=n=nYj=1jt)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj1=n=expZKlogjt)]TJ/F27 11.955 Tf 11.955 0 Td[(zjdnz.2.6>explogCapK+;t2)]TJ/F27 11.955 Tf 7.314 0 Td[(;n>N1:WenowndanupperboundforjPnzj1=n,z2K.SinceKisregular,wecanndacontour)]TJ/F25 7.97 Tf 59.666 -1.794 Td[(1aroundKsuchthaton)]TJ/F25 7.97 Tf 75.339 -1.794 Td[(1wehaveZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt
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SincentendstoKinweakstarsense,on)]TJ/F25 7.97 Tf 123.268 -1.794 Td[(1wehaveuniformlylimn!11 nlogjPnzj=ZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt;Fromthesetwostatementswegetlimsupn!1kPnk1=n)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(1CapKe=2;andhencebythemaximummodulustheorem,limsupn!1kPnk1=nKCapKe=2:.2.7With.2.6and.2.7wearenowreadytocompleteourproof.LetN2besuchthatforn>N2theinequalityin.2.7holds.Ifn>maxfN1;N2g,.2.6and.2.7implythatPnz Pnt1=nmaxfN1;N2gZ)]TJ/F27 11.955 Tf 14.871 18.881 Td[(ftPnz t)]TJ/F27 11.955 Tf 11.955 0 Td[(zPntdtZ)]TJ/F27 11.955 Tf 8.974 18.88 Td[(M Pnz Pnt1=n!ndjtj\051M exp)]TJ/F27 11.955 Tf 9.298 0 Td[(n=2;where\051isthelengthof.Sincethislasttermtendsto0asn!1,thiscompletesourproof. 3.3Proofofiv=iiWeassumethatndoesnotconvergeweakstartoK,andshowthatthereexistsan2suchthatforfz=1=z)]TJ/F27 11.955 Tf 11.955 0 Td[(,Lnfdoesnotconvergeuniformlytof.WerstnotethatiffngdoesnotconvergeweakstartoKthenbyHelly'sselectiontheorem,wecanndasubsequenceoffnsgthatconvergesweakstartosomemeasure6=K.Forconvenience,letusrelabelthissubsequenceasfng.SincenliesontheouterboundaryofK,sodoes.WenextnotethatbyLemma2.1.3ft)]TJ/F27 11.955 Tf 11.956 0 Td[(Lnft=[z1;z2;:::;::zn;t]fnYj=1t)]TJ/F27 11.955 Tf 11.955 0 Td[(zj27

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Thusourproofwillbecompleteifwendan2andasequenceftngK,suchthatforfz=1=z)]TJ/F27 11.955 Tf 11.955 0 Td[(wehave[z1;z2;:::;::zn;tn]fnYj=1tn)]TJ/F27 11.955 Tf 11.956 0 Td[(zj90:.3.8Wewillusethementionedaboveandourlemmasonpotentialtheorytondtheproperaswellasasequenceftngsothat.3.8holds.Werstnd.WebeginbynotingthatbyLemma2.1.2,forfz=1=z)]TJ/F27 11.955 Tf 12.217 0 Td[(,wehave[z1;z2;:::;zn]f=)]TJ/F19 11.955 Tf 9.298 0 Td[(1n+1 Qnj=1zj)]TJ/F27 11.955 Tf 11.955 0 Td[(;sothatj[z1;z2;:::;zn]fj=1 Qnj=1jzj)]TJ/F27 11.955 Tf 11.956 0 Td[(j1=nn=1 exp)]TJ/F27 11.955 Tf 5.48 -9.684 Td[(nRKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(jdnz:Thisalsoshowsj[z1;z2;:::;zn;tn]fj=1 jtn)]TJ/F27 11.955 Tf 11.955 0 Td[(jexp)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(nRKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(jdnz:.3.9AnothereasycalculationgiveusthatnYj=1jtn)]TJ/F27 11.955 Tf 11.956 0 Td[(zjj=0@nYj=1jtn)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj!1=n1An=expnZKlogjtn)]TJ/F27 11.955 Tf 11.956 0 Td[(zjdnz:.3.10Inspectionof.3.8,.3.9,and.3.10nowtellsuswewillbedoneifwecanndanandasequenceftngKsuchthatexp)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(nRKlogjtn)]TJ/F27 11.955 Tf 11.955 0 Td[(zjdnz jtn)]TJ/F27 11.955 Tf 11.955 0 Td[(jexp)]TJ/F27 11.955 Tf 5.48 -9.684 Td[(nRKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(jdnz90:Butwearegoingtochooseoursothat2Kc,andourftngsothatftngK.Wewillthushavej)]TJ/F27 11.955 Tf 11.955 0 Td[(tnjdistancef;Kg>0:Itfollowsthatwewillbedoneassoonaswend2,ftngKsuchthatexp)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(nRKlogjtn)]TJ/F27 11.955 Tf 11.955 0 Td[(zjdnz exp)]TJ/F27 11.955 Tf 5.479 -9.683 Td[(nRKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(jdnz90:.3.11Nowsincen6=K,byLemma2.4.2,wecanalwaysndatnsuchthatZKlogjtn)]TJ/F27 11.955 Tf 11.955 0 Td[(zjdnzlogCapK:28

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Wewillthushaveshown.3.11ifwecanndsuchthatZKlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(jdnz
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4LebesgueConstantsonLejaPointson[-1,1]Inthissectionweshallpresentthemodelforourmainresult.Namely,wewillprovethefollowingTheorem:Theorem4.0.1LetfxjgbeasequenceofLejapointson[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1].ThentheLebesgueconstantsassociatedwiththeseLejapointsaresubexponential,i.e.limn!11=nn=1:However,sincetheproofisratherlong,itwillbeconvenienttodivideitintotwoparts.TherstpartofourproofwillbecontainedinoursubsectionentitledReductionofthemainproof,anditwillbeginwithasimpleobservationthatwhenconsideringthenthrootsofn,oneactuallyneedonlytoconsiderthequotientofacertainpolynomialandacertainproduct.Followingthisobservationwewillcitearesultfrompotentialtheorytellingusthatthepolynomialinvolvedinourquotientcanbeentirelyhandledwithpotentialtheory.Wewillconcludepartonebyshowingthattheproductinvolvedinourquotientcanalmostbehandledentirelywithpotentialtheory.Thusthissubsection,inessence,willshowthatpotentialtheoryalmostgivesusourentireresult.Parttwoofourproofwillconsistofdealingwiththesmallportionofourproofthatpotentialtheorydoesnothelpuswith.However,thissmallportionwillbequitediculttohandle.AproductoftheformYjxj)]TJ/F28 7.97 Tf 6.586 0 Td[(xkj
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Theorem4.1.1Bernstein'sinequalityLetPnbeapolynomialofdegreen.Thenfort2[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]wehavethefollowinginequalityjP0ntjn p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2kPnk[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]:Seee.g.[3].Theorem4.1.2Markov'sinequalityLetPnbeapolynomialofdegreen.Thenfort2[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1]wehavethefollowinginequalityjP0ntjn2kPnk[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]:Seee.g.[3].Lemma4.1.3LetfxjgbeasequenceofLejapointson[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1]andletn>j.Thenxn0andxj0implythatjxn)]TJ/F27 11.955 Tf 11.955 0 Td[(xjjp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xn+p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xj 2n:Proof.LetPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1x=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Yi=1x)]TJ/F27 11.955 Tf 11.955 0 Td[(xi;andMn)]TJ/F25 7.97 Tf 6.586 0 Td[(1=kPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1k[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]:ThenbyBernstein'sinequalitywehavejP0n)]TJ/F25 7.97 Tf 6.587 0 Td[(1tjn)]TJ/F19 11.955 Tf 11.955 0 Td[(1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2Mn)]TJ/F25 7.97 Tf 6.586 0 Td[(1n p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(tMn)]TJ/F25 7.97 Tf 6.587 0 Td[(1:SincexnisthenthLejapoint,wehaveMn)]TJ/F25 7.97 Tf 6.587 0 Td[(1=jPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1xnj.Also,letxj0,1jn,begivenweareassumingxn0.Thenwehavethefollowing:Mn)]TJ/F25 7.97 Tf 6.587 0 Td[(1=jPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1xnj=ZxnxjP0n)]TJ/F25 7.97 Tf 6.586 0 Td[(1tdtnMn)]TJ/F25 7.97 Tf 6.587 0 Td[(1Zxnxjdt p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t=nMn)]TJ/F25 7.97 Tf 6.587 0 Td[(12p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xn)]TJ/F32 11.955 Tf 11.955 9.997 Td[(p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xj=2nMn)]TJ/F25 7.97 Tf 6.586 0 Td[(1jxn)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xj+p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xnBylookingattherstandlasttermsabove,itfollowsthatjxn)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj1 2p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xn+p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xj n;whichcompletestheproof. Lemma4.1.4LetfxjgbeasequenceofLejapointson[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1]andletn>j.Thenjxn)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj1=n2.31

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Proof.WeagainletPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1x=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Yi=1x)]TJ/F27 11.955 Tf 11.955 0 Td[(xi:ThenMarkov'sinequalitytellsusthatP0n)]TJ/F25 7.97 Tf 6.587 0 Td[(1tn2kPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1k[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]:Thisinequalitycombinedwiththemeanvaluetheoremgivesus:jPn)]TJ/F25 7.97 Tf 6.586 0 Td[(1xj)]TJ/F27 11.955 Tf 11.955 0 Td[(Pn)]TJ/F25 7.97 Tf 6.586 0 Td[(1xnj jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xnjn2kPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1jj[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]:SincePn)]TJ/F25 7.97 Tf 6.587 0 Td[(1xj=0,andsincebythedenitionofthenthLejapointwealsohavePn)]TJ/F25 7.97 Tf 6.586 0 Td[(1xn=kPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1k[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1],itfollowsthatkPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1k[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1] jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xnjn2kPn)]TJ/F25 7.97 Tf 6.587 0 Td[(1k[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1];fromwheretheclaimfollows. Lemma4.1.5LetfxjgbeasequenceofLejapointson[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1],leti;jnandletnx=p 1)-222(jxj n+1 n2Thenxi0andxj0implythatjxi)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj1 4nxi+nxjProof.Leti;jn,besuchthatxi0andxj0,andassumewithoutlossofgeneralitythati
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Sincejn,thisimpliesjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xij1 4p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xj n+1 n2!+1 4p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(xi n+1 n2;asweclaimed. 4.2ReductionoftheMainProofWearenowreadytobeginourproof.Asstatedintheintroductiontothissection,theproofwillbedividedintotwoparts.Thisispartone.LetfxjgbeasequenceofLejapoints.Wemustshowthatlimn!11=nn=1;.2.1wheren=supx2[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]nXj=1Qnj=1;j6=kjx)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj Qnj=1;j6=kjxk)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj!.2.2However,wewishtondasimplerexpressionthan.2.2.Withthisinmind,wemakethefollowingobservation:1supx2[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]nXj=1Qnj=1;j6=kjx)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj Qnj=1;j6=kjxk)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj!nmaxk=1;:::;nkPn;kk[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1] Qn;k;wherePn;kx=nYj=1;j6=kx)]TJ/F27 11.955 Tf 11.955 0 Td[(xj;andwhereYn;k=nYj=1;j6=kjxk)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj:Sincetheleftinequalitygives1n,inordertoshow.2.1itsucestoshownmaxk=1;:::;nkPn;kk[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1] Qn;k1=n!1:Butn1=n!1,andbyTheorem2.4.11wealreadyknowthatasn!1,kPn;kk1=n[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1]!1=2uniformlyink.Thusinordertoverify4.2.1,itactuallysucestoprovethatYn;k1=n!1=233

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uniformlyinkasn!1.WepauseheretonotethatshowingthatthenthrootofQn;kconvergesuni-formlyinkto1=2isnotsoeasyasshowingtheconvergencefornitelymanyvaluesofk.Thisisbecausekrunsfrom1ton.However,justasuniformitycameeasilyintheproofof)]TJ/F20 11.955 Tf 5.48 -9.684 Td[(kPn;kk[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]1=n!1=2;uniformitywillalsocomeintheproofofYn;k1=n!1=2:ToprovethisconvergencewebeginbydeningP1n;k;=Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkjjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj;andP2n;k;=Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkj0thereexistsaandanNsuchthat1=2)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkjjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n<1=2+fork=1;:::;nwhenevern>N.34

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BForall>0thereexistsaandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkjN.ButstatementAisthecontentofTheorem2.4.12notethatthecapacityof[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]is1/2,see[9].ThereforetocompletetheproofweonlyhavetoshowstatementB.Wehavenowcompletedtherstpartofourproof.WehavereducedourproblemtoshowingstatementB.WewillhandlestatementBinparttwoofourproof.Note:thereisnoupperboundgiveninstatementBbecauseweareassumingwehave<1andhencetheproductinquestionis1.ThusparttwoofourproofwillconsistofndingalowerboundforthenthrootofP2n;k;.Comments:Themainideaoftheproofisessentiallythatwhenndingthelimitof1=nnasntendstoinnity,oneonlyneedstoconsiderthenthrootsofquotientsoftheformkPn;kk[)]TJ/F25 7.97 Tf 6.587 0 Td[(1;1] Qnj=1;j6=kjxk)]TJ/F27 11.955 Tf 11.956 0 Td[(xjj;andthatpotentialtheoryalmostgivesustheentireresult.Thatistosay,potentialtheorygivesusthatthenthrootsofbothkPn;kk[)]TJ/F25 7.97 Tf 6.586 0 Td[(1;1]andP1n;k;convergeuniformlyink.Nowthereasonforourrepetitionhereinthesecomments,aswellasthereasonforourseparationoftheproofintotwosteps,istostressthatpotentialtheoryalmostgivesusthewholeresult.ItisonlyP2n;k;thatpotentialtheorydoesnothelpuswith,anditisinsteptwothatwewilldealwiththisproduct.Intheabsenceoftheoremsfrompotentialtheory,insteptwowewilluseourseparationlemmaregardingtheLejapointstoattainourdesiredresultsregardingP2n;k;.4.3EstimateforTheMainProductInthisstepwecompletetheproofthat1=nn!1byshowingthatBForall>0thereexistsaandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjxj)]TJ/F28 7.97 Tf 6.586 0 Td[(xkjN.Nowthisisnotatalleasytoprove,andwewillneedtobecleverinourapproach.Webeginbyassumingthattheleftneighborhoodofxkiscontainedin[0;1],thatistosay,webeginbyassumingthat0xk)]TJ/F27 11.955 Tf 12.206 0 Td[(
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foursets:X1=fxj:jn;xkxj1+xk 2g;X2=fxj:jn;1+xk 20thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yxj2Xijxj)]TJ/F27 11.955 Tf 11.956 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N,fori=1;2;3;4:Weclaimthatthisisenoughtocompletetheproof.Indeed,com-pletingthesefoursubstepswillclearlyshowthattheinequalityinBholdsforthoseksatisfyingourassumption,andtheonlyreasonforassumingthat0xk)]TJ/F27 11.955 Tf 12.243 0 Td[(istoguaranteethatxj0forallthexj'sinoursetsXi's.Andtheonlyreasonforneedingxj0,issothatwecanapplyLemma4.1.5whenestimatingjxk)]TJ/F27 11.955 Tf 11.048 0 Td[(xjj.ButtheexactsameinequalityusedinLemma4.1.5appliestonegativexj's,namelythatjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjp 1)-222(jxjj n+1 n2+p 1)-222(jxkj n+1 n2:Thusinprovingthecasewhen0xk)]TJ/F27 11.955 Tf 12.447 0 Td[(,wewillhavealsoproventhecasewhenxk+0.Nowiftheneighborhoodofxkcontains0,wesimplyreplacexkwithitsclosestxj,wherexj<0andformtheproduct0@Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xjj<;xj<0jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj1A1=n0@Yjxk)]TJ/F28 7.97 Tf 6.587 0 Td[(xjj<;xj0jxk)]TJ/F27 11.955 Tf 11.956 0 Td[(xjj1A1=njxk)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj1=n.3.336

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hereweareassumingxk0.Nowbytheabovediscussion,eachoneofthersttwotermsinthisproductwilltendto1.Thelasttermwilltendto1sincebyLemma4.1.4wealwayshavejxk)]TJ/F27 11.955 Tf 12.292 0 Td[(xjj>1=n2.AndsincetheproductofthesethreetermsformsalowerboundforP2n;k;wedonotneedtondanupperboundofcourse,wewillhaveshownthatP2n;k;approximates1forsucientlysmall,andsucientlylargeN.Thuswewillhavecompletedourproofbycompletingthesefoursubsteps.Beforecompletingthesefoursubstepswemakethenalcommentthatwewillassumethatxk6=1.Wemaydothissinceifxk=1,wecansimplyreplacexkwithitsclosestxj,andjustasin.3.3,theproductwilltendto1.WerstshowthatForall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yxj2X1jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N.Thatis,wewillndalowerboundfor0@Yxj2X1jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n;andshowthatthislowerboundtendsto1astendsto0.TodothiswebeginbynotingthattheasymptoticdistributionofLejapointsistheequilibriumdistribution,hence,byLemma2.4.5,ifm1=m1nisthetotalnumberofLejapointsxjjncontainedinX1,thenm1
PAGE 42

Fromthisitthenfollowsthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk 2=1+xk 2)]TJ/F27 11.955 Tf 11.956 0 Td[(xkjxjm1)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjm1 4nxkm1 4p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk n:Lookingattherstandlasttermsintheabovestring,weobtainp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xkm1 2n:Wethenhavejxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjs 4nxks 4p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk ns 4m1 2n1 n=sm1 8n2:Fromthisweobtainm1Ys=1jxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjm1!mm11 n2m1m2m11exp)]TJ/F27 11.955 Tf 9.299 0 Td[(m1 p 8n2m1m2m11 p 8en2m1n2n p 8en2n= p 8e2n:Notethatinthesecondinequalityaboveweusedthatn!n en;whichcanbeeasilyprovenbyinductiononnforlargenitalsoeasilyfollowsfromStirlingsformula,i.e.,n! p 2nnexp)]TJ/F27 11.955 Tf 9.298 0 Td[(n!1;asn!1;andinthelastinequality,weusedthatthefunctionfx=x p 8en2xisdecreasingon[1;n]andthatm1n.Thisprovesthatm1Ys=1jxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj!1=n p 8e2;andthisiswhatwewantedtoshow,because,as!0,wehave!0,andso p 8e2!1: Wenextshowthat38

PAGE 43

Forall>0thereexistsaandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yxj2X2jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N.Thatis,wewillndalowerboundfor0@Yxj2X2jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n;andshowthatthislowerboundtendsto1astendsto0.TodothiswebeginbynotingthatbyLemma2.4.5,ifm2=m2nisthetotalnumberofLejapointsxjjncontainedinX2,thenm2
PAGE 44

=m2 4n2m2+1n 4n2n+1= 42n+1 42n+n= 44nNotethatinthethirdinequalityweusedthefactthatfx=x 4n2x+1ismonotonedecreasingon[1;n]andthat1m20thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yxj2X3jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N.Thatis,wewillndalowerboundfor0@Yxj2X3jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n;andshowthatthislowerboundtendsto1astendsto0.TodothiswebeginbynotingthatbyLemma2.4.5,ifm3=m3nisthetotalnumberofLejapointsxjjncontainedinX3,thenm3
PAGE 45

Wethenhavejxjs+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjsj1 4)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(nxjs+1+nxjs1 42nxjs1 2nxk:WenotethatintherstinequalityaboveweusedLemma4.1.5.Thesecondandthirdinequalitiesfollowfromthefactthatnxisdecreasingon[0;1].Sowehavejxjs+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjsj1 2nxk;andthusjxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj=jxjs)]TJ/F27 11.955 Tf 11.956 0 Td[(xjs)]TJ/F26 5.978 Tf 5.756 0 Td[(1j+jxjs)]TJ/F26 5.978 Tf 5.756 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjs)]TJ/F26 5.978 Tf 5.756 0 Td[(2j+:::+jxj1)]TJ/F27 11.955 Tf 11.956 0 Td[(xkjs 2nxk:Wenowrecallthatweareassumingxk<1.Since1)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(xk
PAGE 46

Notethatintheabovecalculations,weusedStirlingsformulainthesecondinequality,andweusedthatfx=x 2en2xismonotonedecreasingon[1;n]inthethirdinequality.Ifwetakethenthrootsoftherstandlasttermsoftheabove,weobtainthat0@Yxj2X3jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n 2e2;andsince 2e2!1as!0,whichisthesameas!0,thisiswhatwewantedtoshow. Finally,weshowthatForall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yxj2X4jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N.Thatis,wewillndalowerboundfor0@Yxj2X4jxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n;andshowthatthislowerboundtendsto1astendsto0.Thiswillbethemostdicultofourfoursubsteps.RecallingthatX4=fxj:xj2[xk)]TJ/F27 11.955 Tf 11.955 0 Td[(;1)]TJ/F19 11.955 Tf 11.956 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(xk]g;webeginbyassumingthat=2R+1)]TJ/F27 11.955 Tf 12.163 0 Td[(xk,andbydividing[xk)]TJ/F27 11.955 Tf 12.163 0 Td[(;1)]TJ/F19 11.955 Tf 12.163 0 Td[(21)]TJ/F27 11.955 Tf 12.162 0 Td[(xk]intotheRintervals[1)]TJ/F19 11.955 Tf 11.955 0 Td[(2r+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk;1)]TJ/F19 11.955 Tf 11.956 0 Td[(2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xk];r=1;2;:::;R:AgaincitingLemma2.4.5,ifwedenotethenumberofLejapointsintherthintervalbykr,wehavePkrn.WewilldenotetherthintervalbyIr,andforthoseintervalswithkr>0,wewilllabeltheLejapointsinincreasingorder:xj1
PAGE 47

forthoseintervalsIrwithkr>0andthatthenthrootofthisproducttendsto1as!0.ButwhatwereallywanttoshowisthathereweconsiderQxj2Irjxj)]TJ/F27 11.955 Tf 11.792 0 Td[(xkjtobe1forthosekrequalto00@RYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n!1uniformlyinkasn!1andas!0.Andsincexkmaybecloseto1,Rmaybeverylarge,itisnotenoughtosimplyshowthat0@Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n!1fornitelymanyvaluesofr.ThusafterobtainingtheresultthatYxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjkr 4n2kr;wewillworktoobtainaresultcomparingthesizeofallkrtoaxedkr.WewillthenestimateourproductRYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjintermsofkr.Wewillcompletetheproofbyshowingthatthenthrootofthislowerboundtendsto1.WebegintoshowthatYxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjkr 4n2kr:Ifxj1;xj2;:::;xjkraretheLejapointsinIr=[1)]TJ/F19 11.955 Tf 11.955 0 Td[(2r+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk;1)]TJ/F19 11.955 Tf 11.956 0 Td[(2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xk];wethenhavethefollowingjxjs+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjsj1 4)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(nxjs+1+nxjs1 2nxjs1 2p 1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 11.956 0 Td[(2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xk n=1 22r=2p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk n:IntherstinequalityintheaboveequationsweusedLemma4.1.5.Inthesecondandthirdinequalitiesweusedthefactthatnxisdecreasingon[0;1].Nowsincejxjs+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjsj2r=2p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(xk 2n;43

PAGE 48

itfollowsthat2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xk=1)]TJ/F19 11.955 Tf 11.956 0 Td[(2r+1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 11.955 0 Td[(2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjxjkr)]TJ/F27 11.955 Tf 11.955 0 Td[(xjkr)]TJ/F26 5.978 Tf 5.757 0 Td[(1j+jxjkr)]TJ/F26 5.978 Tf 5.756 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjkr)]TJ/F26 5.978 Tf 5.756 0 Td[(2j+:::+jxj2)]TJ/F27 11.955 Tf 11.955 0 Td[(xj1jkr 42r=2p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk n:Itiseasytoseethatthesameinequalityistruealsoforkr=1.Dividingtherstandlasttermsintheabovestringofinequalitiesby2rp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xkagainrecallthatweareassumingxk6=1,weobtainkr2)]TJ/F28 7.97 Tf 6.586 0 Td[(r=2 4np 1)]TJ/F27 11.955 Tf 11.956 0 Td[(xk:.3.5Bysquaringbothsidesweobtaink2r2)]TJ/F28 7.97 Tf 6.586 0 Td[(r 16n21)]TJ/F27 11.955 Tf 11.955 0 Td[(xk:Ifwenowtakethisandmultiplybothsidesby2rweobtain2r)]TJ/F27 11.955 Tf 11.955 0 Td[(xkk2r 16n2:Wearenowreadytoconsidertheproductofourjxjs)]TJ/F27 11.955 Tf 11.897 0 Td[(xkj'sforthoseintervalsIrwithkr6=0.WehavekrYs=1jxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjr)]TJ/F27 11.955 Tf 11.955 0 Td[(xkkrk2r 16n2kr=kr 4n2kr:.3.6Sincekrn,wehavekrYs=1jxjs)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj!1=n 42!1;as!0:AtthispointwewouldhavecompletedourproofhadwebeendealingwithanitexedR.Buttorepeatwhatwasstatedabove,whatwereallywanttoshowisthat0@r=RYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n!1uniformlyinkasn!1,!0andwhereQxj2Irjxj)]TJ/F27 11.955 Tf 12.42 0 Td[(xkjisconsideredtobe1whenkr=0.Withthehelpof4.3.5wewillnowndakr;CsuchthatkrCkr,andwiththehelpof.3.6wewillthencompleteourproof.Tondkr;C,webeginby44

PAGE 49

recallingthatifnistheprobabilitymeasureconcentratedontherstnLejapoints,withthevalueof1=nateachpoint,thenn*seepreliminaries.FromthisbyLemma2.4.9itfollowsthatthereexistsNsuchthatN1 4Z1)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F28 7.97 Tf 6.587 0 Td[(xk1)]TJ/F25 7.97 Tf 6.586 0 Td[(2R)]TJ/F28 7.97 Tf 6.587 0 Td[(xk1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t2dt:.3.7Nowsince1 p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(t21 21 p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(t;wecanrewrite.3.7asRXr=1kr n>1 4Z1)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F28 7.97 Tf 6.587 0 Td[(xk1)]TJ/F25 7.97 Tf 6.586 0 Td[(2R)]TJ/F28 7.97 Tf 6.587 0 Td[(xk1 p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(tdt:ByintegratingweobtainRXr=1kr n>1 2p 2R)]TJ/F27 11.955 Tf 11.955 0 Td[(xk)]TJ/F19 11.955 Tf 16.685 8.088 Td[(1 2p 21)]TJ/F27 11.955 Tf 11.955 0 Td[(xk=1 2p 2Rp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk1)]TJ 16.613 17.978 Td[(p 2 p 2R!:AfterthisroughestimatewerewritethisinequalityasRXr=1kr n>1 2p 2Rp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk2 7=1 7p 2Rp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk:.3.8Wearenowreadytondourkr.Withthehelpof.3.5and.3.8wewillndalargekr,andwewilllabelthislargekraskr.TodothiswenotethatforlargeMandforR>M,wehavethefollowing4p 2)]TJ 11.956 9.89 Td[(p 2R)]TJ/F28 7.97 Tf 6.586 0 Td[(M 1)]TJ 11.955 9.889 Td[(p 2!p 2R1 31 7.3.9WexsuchanMactually,M=20wouldsuce.WenowclaimthatwemusthavemaxkR n;kR)]TJ/F25 7.97 Tf 6.587 0 Td[(1 n;:::;kR)]TJ/F28 7.97 Tf 6.586 0 Td[(M n1 Mp 2R1 31 7p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(xk:Indeed,ifthisisnotthecasethenby.3.5and.3.9,wehavethefollowingRXr=1kr n=R)]TJ/F28 7.97 Tf 6.586 0 Td[(M)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xr=1kr n+RXr=R)]TJ/F28 7.97 Tf 6.587 0 Td[(Mkr nR)]TJ/F28 7.97 Tf 6.587 0 Td[(M)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xr=14p 2rp 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk+RXr=R)]TJ/F28 7.97 Tf 6.587 0 Td[(Mkr n=45

PAGE 50

4p 2)]TJ 11.956 9.889 Td[(p 2R)]TJ/F28 7.97 Tf 6.586 0 Td[(M)]TJ/F25 7.97 Tf 6.586 0 Td[(1 1)]TJ 11.955 9.89 Td[(p 2!p 1)]TJ/F27 11.955 Tf 11.956 0 Td[(xk+RXr=R)]TJ/F28 7.97 Tf 6.586 0 Td[(Mkr np 2R1 31 7p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xk+RXr=R)]TJ/F28 7.97 Tf 6.587 0 Td[(Mkr n

N,byshowingthat0@RYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n46

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approximates1forsucientlysmallandn>N.Nowtoshowthelatter,asintheothersubsteps,wewillndalowerboundfor0@RYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n;andshowthatthislowerboundtendsto1forlargeN.WeproceedasfollowsRYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkjRYr=1kr 4n2krRYr=1kr 4en2krRYr=1kr2r)]TJ/F29 5.978 Tf 5.756 0 Td[(R 2 4en!22r)]TJ/F29 5.978 Tf 5.756 0 Td[(R 2krWenotethatintherstinequalityweused4.3.6,andinthesecondinequalityweusedthatfx=k xkisdecreasingon0;1.Inthethirdinequalityweused.3.11,andthatfx=x 4enxisdecreasingon[1;2n].Notethatkr2r)]TJ/F29 5.978 Tf 5.756 0 Td[(R 2<2nsincekr,thesecondproductintherighthandsideof.3.12isgreaterthanthekr-thpowerofsomepositiveconstantQ.WecanthuscontinuewithRYr=1kr2r)]TJ/F29 5.978 Tf 5.757 0 Td[(R 2 4en!22r)]TJ/F29 5.978 Tf 5.756 0 Td[(R 2krRYr=1kr 4en22r)]TJ/F29 5.978 Tf 5.756 0 Td[(R 2krQkr=RYr=1kr 4en2kr!p 2r)]TJ/F29 5.978 Tf 5.756 0 Td[(RQkr=kr 4en2kr!p 2)]TJ/F25 7.97 Tf 5.756 -0.498 Td[(p 21)]TJ/F29 5.978 Tf 5.756 0 Td[(R p 2)]TJ/F26 5.978 Tf 5.756 0 Td[(1Qkrn 4en2np 2)]TJ/F25 7.97 Tf 5.756 -0.498 Td[(p 21)]TJ/F29 5.978 Tf 5.756 0 Td[(R p 2)]TJ/F26 5.978 Tf 5.756 0 Td[(1Qn:47

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Takingnthrootsoftherstandlasttermsinourstringofinequalities,weobtain0@RYr=1Yxj2Irjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=n 4e2p 2 p 2)]TJ/F26 5.978 Tf 5.756 0 Td[(1Q:Sincethislasttermtendsto1as!0,whichisthesameas!0,thiscompletesthefourthsubstepandthustheentireproof. Actually,intheprecedingproof,wedidmorethanprovethattheLebesguecon-stantsassociatedwithLejasequenceson[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1]aresubexponential.SinceinourproofweonlyneededthatthemeasuresassociatedwithLejasequencesconvergeweakstartotheequilibriumdistributionfor[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1],andthatanytwopointsxj;xninoursequencesatisfythedistancingrulejxj)]TJ/F27 11.955 Tf 11.956 0 Td[(xnj1 4p 1)-222(jxjj n+1 n2!+1 4p 1)-222(jxjj n+1 n2!;j
PAGE 53

5LejaPointsonMoreGeneralSetsWenowconsiderLejapointsonmoregeneralsets.WewillrstextendourresultconcerningtheLebesgueconstantsassociatedwithLejapointson[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]Theorem4.0.1toacurveinthecomplexplane.Motivatedbythisextension,wewillthenproceedtogivearesultconcerningtheunionofnitelymanynicecompactsets.5.1LejaPointsonanArcRecallthataJordanarcisthehomeomorphicimageof[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1].InextendingTheorem4.0.1toarcsinthecomplexplane,weshalluseasaguidetheproofofTheorem4.0.1.ThereadershouldrecallthatpotentialtheoryalmostgaveusthewholeproofofTheorem4.0.1.Thatis,wereducedourproblemgreatlywiththehelpofTheorem2.4.11andTheorem2.4.12.ButTheorems2.4.11and2.4.12applytocurvesintheplaneaswellastheinterval[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1],andthuswewillagainbeabletoreduceourproblemgreatlywiththeaidofPotentialtheory.ThereadershouldalsorecallthatinthesectionofTheorem4.0.1'sproofentitledestimateforthemainproduct,themainideausedwasadistancingruleforLejapointson[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1].WewillalsoneedadistancingruleforourproofregardingLejapointsonacurve.Now,inthecaseof[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]weobtainedourdistancingrulethroughapplicationsofBernstein'sinequalityandMarkov'sinequality.Similarly,inourproofregardingLejapointsonacurve,weshallapplyacomplexversionofBernstein'sinequalityaswellasMarkov'sinequality.However,inobtainingourversionofBernstein'sinequality,weshallneedtheBernstein-Walshlemma.OurapplicationoftheBernstein-WalshLemmawillrequireustorestrictourselvestocurveswhichareC1andwhichhavenonzeroderivatives.ThereasonforneedingtheassumptionthatourcurveisC1,issothatwewillhaveatourdisposalthefollowingresultofDzjadyk[4].ItspeaksaboutthelevelsetsoftheGreenfunctionassociatedwithasmoothJordanarc.Ingeneral,theGreen'sfunctionassociatewithacompactsetKofpositivecapacityisdenedasgz=Zlogjz)]TJ/F27 11.955 Tf 11.955 0 Td[(tjdKt)]TJ/F19 11.955 Tf 11.955 0 Td[(logCapK;whereKistheequilibriummeasureofK.Theorem5.1.1Let)]TJ/F44 11.955 Tf 12.04 0 Td[(beaJordanarcintheplanewithcontinuouscurvature,andwithendpointsatz06=1andat1,andletgbetheGreenfunctionassociatedwith)]TJ/F44 11.955 Tf 7.314 0 Td[(.49

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Letzbeapointon)]TJ/F44 11.955 Tf 11.364 0 Td[(whichiscloserto1thantoz0.Thenthedistance,z,fromztothelevelcurveg=1=nsatisesthefollowinginequalityzC np j1)]TJ/F27 11.955 Tf 11.955 0 Td[(zj+1=n2whereCisaconstantdependingonlyonthecurve.Comment:InDzjadyk'spaper[4],hisresultwasmoregeneral.Hisresultconcernedpiecewisesmoothcurvesandtheirjunctionpoints,ratherthanmerelycurveswithendpointat1.Wehavehowever,restrictedourselvestotheveryspecialcaseofhisresultwhichweshallneed.Asstatedabove,weneedthisinourapplicationoftheBernstein-Walshlemma,whichwenowstateforaproofsee[12].Theorem5.1.2Bernstein-WalshLemmaLetKbeacompactsubsetoftheplaneandgitsassociateGreenfunction.ThenjPnzjkPnkKexpngz;z2C:WiththeBernstein-WalshlemmawearenowreadytogiveourcomplexversionoftheBernsteininequality.Lemma5.1.3LetPnbeapolynomialofdegreenandlet)]TJ/F44 11.955 Tf 11.714 0 Td[(beanopencurveinthecomplexplanewithendpointsatz06=1andat1.Thenforallpointsz2)]TJ/F44 11.955 Tf 11.639 0 Td[(whichliecloserto1thantoz0,thefollowinginequalityholdsjP0nzjnCkPnk)]TJET1 0 0 1 327.281 355.541 cmq[]0 d0 J0.478 w0 0.239 m47.145 0.239 lSQ1 0 0 1 -327.281 -355.541 cmBT/F32 11.955 Tf 328.338 353.15 Td[(p j1)]TJ/F27 11.955 Tf 11.956 0 Td[(zj:Proof.AsinTheorem5.1.1,letzbethedistancefromztothelevelcurveg=1=n,wheregistheGreenfunctionassociatedwith)1(.Thensincezlieson,allpointstwithinzofzarealsowithinthelevelcurveg=1=n.Thusforallsuchtwehavegt1=n.ByCauchy'sTheorem,theBernstein-WalshLemma,andTheorem5.1.1wethenhavejP0nzj=1 2iZjt)]TJ/F28 7.97 Tf 6.586 0 Td[(zj=zPnt t)]TJ/F27 11.955 Tf 11.956 0 Td[(z2dt1 zsupjt)]TJ/F28 7.97 Tf 6.586 0 Td[(zj=zjPntj1 zkPnk)]TJ/F19 11.955 Tf 7.779 1.793 Td[(expnsupjt)]TJ/F28 7.97 Tf 6.586 0 Td[(zj=zgt1 zkPnk)]TJ/F19 11.955 Tf 7.779 1.793 Td[(expn=nnCkPnk)]TJ/F19 11.955 Tf 7.78 1.793 Td[(exp p j1)]TJ/F27 11.955 Tf 11.956 0 Td[(zj+1=n2nCkPnk)]TJET1 0 0 1 315.939 144.337 cmq[]0 d0 J0.478 w0 0.239 m47.145 0.239 lSQ1 0 0 1 -315.939 -144.337 cmBT/F32 11.955 Tf 316.995 141.946 Td[(p j1)]TJ/F27 11.955 Tf 11.955 0 Td[(zj;aswedesired.wenotethatinthelastinequalityCabsorbedexp 50

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WhileBernstein'sinequalityforcurvesrequiresustointroduceDzjadyk'sresultonthelevellinesofGreenfunctionsandisslightlymoreinvolvedthaninthe[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]case,Markov'sinequalityforanarcoractuallyforanyconnectedcompactsetinthecomplexplaneisalmostidenticaltothe[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]case.Theorem5.1.4Markov'sinequalityLetPnbeapolynomialofdegreen,andlet)]TJ/F44 11.955 Tf 11.422 0 Td[(beaJordanarcinthecomplexplane.ThenthereexistsaconstantCsuchthatthefollowinginequalityholdsjP0nzjCn2kPnk)]TJ/F27 11.955 Tf 5.786 1.793 Td[(:Foraproofsee[8].CommentActually,thisisMarkov'sinequalityforcompactsubsetsoftherealline.Weearlierhowever,consideredthecasewhereK=[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1],forwhichtheconstantinMarkov'sinequalityis1.WearenowclosertobeingabletondourdistancingruleforLejapointsonacurve.Thereisstillhowever,onelastpropertyofcurveswhichwewishtodiscuss.Theproofofourdistancingrulewillgomoresmoothlyifwehaveacertainmonotonicityruletoourcurve.Thismonotonicityrulefollowseasilyfromourassumptionsthat)]TJ -193.083 -13.948 Td[(isC1withnonzeroderivative.Lemma5.1.5Let)-413(=xt+iytt2[0;1]beC1arcwithnon-zeroderivative.ThenthereexistssuchthateverydiscDofradiuscenteredatsomez=xtz+iytzhasthepropertythatanytwopointsz1;z22)]TJ/F20 11.955 Tf 15.378 0 Td[(Dcanbeconnectedbyacurve)]TJ/F25 7.97 Tf 7.314 -1.859 Td[([tz1;tz2])]TJ/F44 11.955 Tf 13.104 0 Td[(suchthateitherjx0tj>1=2jy0tjforallt2[tz1;tz2];orjy0tj>1=2jx0tjforallt2[tz1;tz2]:Proof.Letz=xtz+iytz2.Theneitherjx0tzjjy0tzjorjy0tzjjx0tzj.Assumew.l.o.g.thatjx0tzjjy0tzj.Since)-332(isC1,thereexistsadiscDzcenteredatzsuchthatforallw=xtw+iytw2Dz\051,jx0twj>1=2jy0twj:WemayassumethatDzisalsosmallenoughsothatanytwopointsw1;w22Dz\051canbeconnectedbyapath)]TJ/F25 7.97 Tf 114.946 -1.859 Td[([tw1;tw2]D.Since)-319(iscompactthereexistsDz1;:::;Dznsuchthattheunionofthesediscscover.Forthisnitesubcoverthereexistssuchthatalldiscsofradiuswithcenterson)-326(lieinoneoftheDzi.Thisisourrequired. WearenowreadytondourdistancingruleforLejapointsonasmootharcinthecomplexplane.Threesimplelemmasaretofollowing.ThethirdoneisourdistancingruleforLejapointsonacurve.Lemma5.1.6Let)-438(=xt+iytbeaC1arcwithnon-zeroderivativeandwithendpointsatz06=1andat1.LetbeasinLemma5.1.5,andletfzjgbeasequenceofLejapointson)]TJ/F44 11.955 Tf 7.314 0 Td[(.ThenifDisadiscofradiuswhosepointsliecloserto1thantoz0,andifjx0tj>1=2jy0tjforallz=xt+iytinD,wehavep j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj+p j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj 3Cnjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj;wheneverj
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Comment:ThisLemmabasicallysaysthatjxj)]TJ/F27 11.955 Tf 13.141 0 Td[(xnjwillbelargewhenthepointszj;znliecloseto1andwhenjx0jislarge.Wewantthereadertonotethattheconditionsinthislemmapresentnorestrictions.First,thisisbecauseoneofeitherjx0jorjy0jmustbegreaterthanorequaltotheotherone.Second,thisisbecausetheselectionofLejapointsfromacompactsetinthecomplexplaneisinvariantwithrespecttotranslationsintheplane,sothatthepositionsoftheendpointsofourarcareirrelevant.Also,theinLemma5.1.5waschosenwithoutregardtotheendpointsof.Proof.Inwhatfollowswewillusethenotationzi=xti+iyti,andPn=Qnj=1jz)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj.LetMn=kPnk,thenbythedenitionofthenthLejapoint,wehaveMn=ZznzjP0ntdt=ZtntjP0n\050t)]TJ/F30 7.97 Tf 16.419 4.936 Td[(0tdt=ZtntjP0n\050tx0t+iy0tdtZtntjjP0n\050tjjx0tjdt+ZtntjjP0n\050tjjy0tjdthereweareusingLemma5:1:5andtheassumptionthatx0>1=2y03ZtntjjP0n\050tjjx0tjdthereweareusingLemma5:1:33ZtntjCnMnx0t q p )]TJ/F27 11.955 Tf 11.955 0 Td[(xt2+y2tdt3ZtntjCnMnx0t q p )]TJ/F27 11.955 Tf 11.955 0 Td[(xt2dt=3ZtntjCnMnx0t p 1)]TJ/F27 11.955 Tf 11.955 0 Td[(xtdt=3CnMnq j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj)]TJ/F32 11.955 Tf 17.933 10.806 Td[(p j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj;whereinthelaststepweusedthateitherx0t>0orx0t<0ontheinterval[tj;tn].NowfromMn3CnMnq j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj)]TJ/F32 11.955 Tf 17.932 10.805 Td[(p j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj3CnMnjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj p j1)]TJ/F27 11.955 Tf 11.956 0 Td[(xjj+p j1)]TJ/F27 11.955 Tf 11.956 0 Td[(xnj;52

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weobtainp j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xjj+p j1)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj 3Cnjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xnj;aswedesired. Lemma5.1.7Let)-278(=xt+iytbeaC1arcwithnon-zeroderivative,andletfzjgbeasequenceofLejapointson)]TJ/F44 11.955 Tf 7.314 0 Td[(,wherezj=xj+iyj.Further,letbeasinLemma5.1.5,andletDbeadiscofradiusforwhichjx0tj>1=2jy0tj,foralltsuchthatxt+iyt2D.ThenthereexistsC0suchthatforj1=2jy0tjforallz=xt+iytinD,thenjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xijCp jxjj+p jxij n+1 n2!:Similarly,ifDisadiscofradiuswhosepointsliecloserto1thantoz0,andifjy0tj>1=2jx0tjforallz=xt+iytinD,thenjyj)]TJ/F27 11.955 Tf 11.955 0 Td[(yijCp jyjj+p jyij n+1 n2!:53

PAGE 58

Proof.CombiningLemmas5.1.6and5.1.7,andassumingi
PAGE 59

ByTheorem2.4.12,weknowthatP1n;k;1=napproximatesCap\051forlargenandsmall.Thus,justasintheproofofTheorem4.0.1,ourproofhasbeenreducedtoshowingthatforsucientlysmall,P2n;k;1=napproximates1uniformlyinkasn!1.Tobeprecise,wehavereducedourprooftoshowingthefollowing*Forall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjzj)]TJ/F28 7.97 Tf 6.587 0 Td[(zkjjzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zkj1A1=nfork=1;:::;nwhenevern>N.AtthispointwemaketheassumptionthatourissucientlysmallastosatisfytheconclusionofLemma5.1.5.Wefurthermaketheassumptions,w.l.o.g.,that)-282(hasanendpointat1,thatzkliescloserto1thanto'sotherendpoint,andthatinthisneighborhoodofzkwehavejx0tj>1=2jy0tj.Indeed,wedonotloseanygeneralitywiththeseassumptionsbecausetheselectionofLejapointsfromacompactsetisinvariantwithrespecttotranslationsandrotations,andbecauseourassumptionsabout)1(t=xt+iytguaranteethateitherjx0tjislargeorjy0tjislarge.Since0@Yjxj)]TJ/F28 7.97 Tf 6.586 0 Td[(xkjjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nformsalowerboundfor0@Yjzj)]TJ/F28 7.97 Tf 6.586 0 Td[(zkjjzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zkj1A1=n;itfollowsthatwewillhavecompletedourproofifwecanshowthefollowingstatement:**Forall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjxj)]TJ/F28 7.97 Tf 6.587 0 Td[(xkjjxj)]TJ/F27 11.955 Tf 11.955 0 Td[(xkj1A1=nfork=1;:::;nwhenevern>N.Ifwenowmakeonenalassumptionw.l.o.g.that)-234(hastangentliney=0at1,andthatourpointsapproach1fromtheleft,thenourjxj)]TJ/F27 11.955 Tf 9.809 0 Td[(xkj'ssatisfytheinequalitygiveninLemma5.1.8,andourxj'saredistributedaccordingtotheequilibriumdistributionfor[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1]near1see[12].Thus**istheexactstatementprovedinTheorem4.0.1.Thuswehavealreadyshown**.Thiscompletestheproof. 55

PAGE 60

5.2LejaPointsonMoreGeneralSetsNowwewishtoconsidertheunionofpiecewisesmootharcs,curves,andcloseddo-mainswithpiecewisesmoothboundary.Lemma5.1.6willsatisfyourneedsinthissection,aswewilluseittoestimatethedistancebetweenLejapointswhichlieonthesamesmootharc.However,wewillalsoneedanestimateforthedistancebetweenLejapointswhichlieonseparatearcs.SinceLemma5.1.7onlyappliestoLejapointslyingonthesamesmootharcwewillneedthefollowingresult,andthedistancingestimatewhichfollows.Lemma5.2.1[14]LetKbeaconnectedcompactset.ThenforeveryD>0thereisaCDsuchthatifPnisapolynomialofdegreeatmostnthenjP0nzjCDn2kPnkK;distz;KD=n2Lemma5.2.2LetKbetheunionofnitelymanysmootharcs,andletfzjgbeasequenceofLejapointsonK.ThenthereexistsCsuchthatC=n2
PAGE 61

*Forall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjzj)]TJ/F28 7.97 Tf 6.587 0 Td[(zkjN.Inordertoprove*,itsucestoprovethefollowingfori=1;:::;m:**Forall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjzj)]TJ/F28 7.97 Tf 6.587 0 Td[(zkj<;zj2)]TJ/F29 5.978 Tf 5.288 -1.215 Td[(ijzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zkj1A1=nfork=1;:::;nwhenevern>N.Nowtoprove**,therearetwocases.Therstcaseisifzk2)]TJ/F28 7.97 Tf 7.314 -1.794 Td[(i.Inthiscase,Lemma5.1.8holdswhenestimatingthedistancefromzktoanyzj2)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(i.Further,thedistributionofpointszj2)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(iwillbelessdensethanifKwereasinglearc.Thatis,if)]TJ/F28 7.97 Tf 34.425 -1.794 Td[(ihasendpointat1andtangentliney=0approaching1fromtheleft,thenthedistributionwillbelessdensethanthearcsinedistribution.Thusinthecaseofzk2)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(i,**followsfromTheorem5.1.9.Thesecondcasecomeswhenzk=2)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(i.Inthiscase,replacezkin**withzj,wherezjistheLejapointon)]TJ/F28 7.97 Tf 113.903 -1.794 Td[(iwhichisclosesttozk.Thenforanypointzj2)]TJ/F28 7.97 Tf 7.314 -1.794 Td[(i,wewillhavejzk)]TJ/F27 11.955 Tf 11.956 0 Td[(zjj1 2jzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj:Nowwithzjinplaceofzk,**willholdpreciselybecauseallthepointsonthecurve)]TJ/F28 7.97 Tf 38.53 -1.793 Td[(isatisfyaninequalityoftheformjzj)]TJ/F27 11.955 Tf 11.956 0 Td[(zjjCdzj;zj;wheredz;wisadistancingfunctionwhichdependsonthenearestendpointalong)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(itozj.Butwehavejzk)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj1 2jzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj;andthusthatjzk)]TJ/F27 11.955 Tf 11.955 0 Td[(zjj1 2Cdzj;zj;forallzjwherej6=j.Itthenfollowsthatwehave:Forall>0thereexistsanandanNsuchthat1)]TJ/F27 11.955 Tf 11.955 0 Td[(<0@Yjzj)]TJ/F28 7.97 Tf 6.587 0 Td[(zkj<;zj2)]TJ/F29 5.978 Tf 5.289 -1.215 Td[(i;j6=jjzj)]TJ/F27 11.955 Tf 11.955 0 Td[(zkj1A1=nfork=1;:::;nwhenevern>N.57

PAGE 62

Sincewealwayshavejzk)]TJ/F27 11.955 Tf 11.955 0 Td[(zjjC2=n2;whereC2isaconstantwhichdependsonKwehavethisbyLemma5.2.2,itfollowsthat**holdswithzkaswell.Thiscompletestheproof. 58

PAGE 63

References[1]L.Brutman,Lebesguefunctionsforpolynomialinterpolation-asurvey,Theher-itageofP.LChebyshev:afestschriftinhonorofthe70thbirthdayofT.J.Rivlin,Ann.Numer.Math.4997,111-127.[2]M.M.ChawlaandM.K.Jain,ErrorEstimatesforGaussQuadratureFormulasforAnalyticFunctions,MathematicsofComputation,22968,82-90.[3]R.A.DeVoreandG.G.Lorentz,ConstructiveApproximation,GrundlehrenderMathematischenWissenschaften303,Springer-Verlag,Berlin,1993.[4]V.K.,Dzjadyk,OntheTheoryofApproximationoffunctionsonclosedsetsofthecomplexplane,ProceedingsoftheSteklovInstituteofMathematics,134975,75-130.[5]W.Gautschi,NumericalAnalysisAnIntroduction,BirkhauserBoston,Boston,1997.[6]E.LevinandB.Shekhtman,Twoproblemsoninterpolation,ConstructiveAp-prox.,11995,513{515.[7]I.P.Natanson,ConstructiveFunctionTheory,3,FrederickUngarPublishingCo.,NewYork,1965.[8]Ch.Pommerenke,Onthederivativeofapolynomial,MichiganMath.J.,6959,373{375.[9]T.Ransford,PotentialTheoryintheComplexPlane,CambridgeUniversityPress,Cambridge,1995.[10]L.Reichel,NewtonInterpolationatLejaPoints,BIT,30990,332{346.[11]W.Rudin,PrinciplesofMathematicalAnalysis,McGrawHill,NewYork,1964.[12]E.B.SaandV.Totik,LogarithmicPotentialswithExternalFields,GrundlehrendermathematischenWissenschaften,316,Springer-Verlag,NewYork/Berlin,1997.[13]J.SzabadosandP.Vertesi,Interpolationoffunctions,WorldScienticPublishingCo.,Inc.,Teaneck,NJ,1990.[14]V.Totik,Christoelfunctionsoncurvesanddomainsmanuscript.59

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[15]L.N.Trefethen,andJ.A.C.Weideman,Tworesultsonpolynomialinterpolationinequallyspacedpoints,J.Approx.Theory,65991,247{260.60

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AbouttheAuthorRodneyTaylorreceivedhisB.A.andM.S.degreesinmathematicsfromtheUniversityofSouthFlorida.HeenteredthePh.D.programattheUniversityofSouthFloridain2003.Whilestudyingforhisdoctoratehetaughtseveralundergraduatecourses.


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