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Christoffel function asymptotics and universality for Szego weights in the complex plane

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Christoffel function asymptotics and universality for Szego weights in the complex plane
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Findley, Elliot M
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Faber
Orthogonal polynomials
Zero-spacing
Ill-posed problems
Potential theory
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Abstract:
ABSTRACT: In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions. We compute the translated asymptotics, the limit as n proceeds to infinity of "n" times the order-n Christoffel function evaluated at "x" plus "a divided by n", and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté's result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.
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Dissertation (Ph.D.)--University of South Florida, 2009.
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Includes bibliographical references.
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by Elliot M. Findley.
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Christoffel function asymptotics and universality for Szego weights in the complex plane
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ABSTRACT: In 1991, A. Mt precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions. We compute the translated asymptotics, the limit as n proceeds to infinity of "n" times the order-n Christoffel function evaluated at "x" plus "a divided by n", and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Mt's result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.
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System requirements: World Wide Web browser and PDF reader.
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Zero-spacing
Ill-posed problems
Potential theory
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ChristoelFunctionAsymptoticsandUniversalityfor Szeg}oWeightsintheComplexPlane by ElliotM.Findley Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:VilmosTotik,Ph.D. ThomasBieske,Ph.D. EvgueniiRakhmanov,Ph.D. BorisSchekhtman,Ph.D. DateofApproval: March31,2009 Keywords:Faber,OrthogonalPolynomials,Zero-Spacing,Ill-PosedProblems, PotentialTheory c Copyright2009,ElliotM.Findley

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Acknowledgements IaminexpressiblygratefulanddeeplyhumbledtohavestudiedunderVilmos Totik.AmongthemostinsightfulmenIhaveeverknown,heiscertainlythekindest. HisfatherlyguidanceandconstantsupporthaveenrichedmylifemorethanIcan articulateandIwillforevertreasurethetimewehavespenttogether.Isincerely thankthemathematicsandphysicsfacultyattheUniversityofSouthFloridafor theirkindinstructionandtheirmanyinvestmentsintheyoungmindswhodepend uponthem.Amongtheirrank,Ihavefoundatoncethemostgraciousandbrilliant peopleIhaveeverknown.Finally,IpouroutmygratitudetotheLordJesusChrist, thearchitectofallwisdomandtruthandthesourceofeveryfortuity.Hehasheard myprayerandanswered.

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TableofContents Abstractiii 1Introduction1 1.1ClassicalApproximation.........................2 1.2Quadrature................................3 1.3Universality................................4 1.4Zero-Spacing...............................5 1.5ABriefHistory..............................6 2Universality9 2.1TranslatedAsymptotics.........................11 2.2Measureson[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]............................22 2.3AsymptoticsforRegularMeasures....................26 2.4Universality................................30 2.5ZeroDistributionofOrthogonalPolynomials..............32 3ChristoelFunctionAsymptoticsonGeneralCurves35 3.1PotentialTheory.............................35 3.2MainResults...............................37 3.3Applications................................40 OrthogonalPolynomials.........................40 OperatorTheoreticFormulationandIll-PosedProblems.......41 3.4Proofs...................................42 i

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References54 AbouttheAuthorEndPage ii

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ChristoffelFunctionAsymptoticsandUniversalityfor Szeg } oWeightsintheComplexPlane ElliotM.Findley Abstract In1991,A.Matepreciselycalculatedtherst-orderasymptoticbehaviorofthe sequenceofChristoelfunctionsassociatedwithSzeg}omeasuresontheunitcircle. Ourprincipalgoalistheabstractionofhisresultintwodirections:Wecomputethe translatedasymptotics,lim n n ;x + a=n ,andobtain,asacorollary,auniversality limitforthefairlybroadclassofSzeg}oweights.Finally,weproveMate'sresultfor measuressupportedonsmoothcurvesintheplane.Ourproofofthelatterderives, inpart,fromapreciseestimateofcertainweightedmeansoftheFaberpolynomials associatedwiththesupportofthemeasure.Finally,weinvestigateavarietyofapplications,includingtwonovelapplicationstoill-posedproblemsinHilbertspaceand themeanergodictheorem. iii

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1Introduction Let beacompactlysupported,Borelmeasureinthecomplexplane.WeareinterestedinthepreciseasymptoticbehavioroftheassociatedsequenceofChristoel functions, f n ;z g 1 n =0 .When issupportedontherealline,theChristoelfunctionsaredenedasfollows: n ;x =inf 1 j P x j 2 Z j P j 2 d; .0.1 wheretheinmumisevaluatedoverallpolynomials P ofdegreeatmost n )]TJ/F19 11.9552 Tf 12.15 0 Td [(1that donotvanishat x .Formeasuressupportedontheunitcircleoranysmoothhomeomorphicimagethereof,theinmumrangesoverthecomplexpolynomialsandthe denitionisslightlymodied: n ;z =inf 1 j P z j 2 1 2 Z j P j 2 d: .0.2 Inmuchoftheliterature,thelatterisdenoted n ;e i when issupportedonthe unitcircle.Wewilladheretothisconventioninchapter2,butreverttothenotation of.0.2inthenalchapter. Frominterpolationandquadraturetostochasticprocessesandstatisticalinference,thissimplesequencehasdiverseutility.Forthesimplestexample,welookto orthogonalpolynomials.Let f p n g n denotethesequenceoforthonormalpolynomials associatedwith .Ifweexpand P x in.0.1intermsofthissequence,thenelemen1

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tarylinearalgebraprovidesthefollowingrepresentationoftheChristoelfunctions: 1 n ;x = n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 j p k x j 2 : .0.3 When issupportedonahomeomorphicimageoftheunitcircle,asin.0.2,we willtake p n orthonormalto = 2 ,sothatthisidentitywillholdforallmeasures, regardlessofthesupport.TherelationshipbetweenChristoelfunctionsandorthogonalpolynomialsisdeep.Indeed,PaulNevai,inhisveryhelpfulsurveyofthe contributionsofGezaFreud[14],easilyderivesSzeg}o'sseminaltheoryoforthogonalpolynomialsontheunitcircleentirelyfromaconsiderationofChristoel-function asymptotics. ForastellarexplorationoffurtherapplicationsofChristoelfunctionasymptotics, wereferthereadertoGrenander'sandSzeg}o'streatiseonToeplitzmatrices[5]. Beforeproceedingtoourmainresults,wewillshowcasetheversatilityofthissubject withafewmoreexamples,thersttakenfromclassicalapproximationtheory. 1.1ClassicalApproximation Considersomeorthonormalbasisofpolynomials f p k g withrespecttothemeasure Wemayapproximateany f 2 L 2 bythefollowingpolynomials: S n = n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 c k p k ; where c k := Z f x p k x d x : S n isalinearfunctionof f ,sowemaydenethelinearoperators L n f := S n and associatednorms n x :=sup k f k L 1 1 j L n f x j ; calledtheLebesguefunctions. f L n g areintegraloperators,withassociatedkernels K n x;t := n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 p k x p k t ; 2

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calledthereproducingkernelsassociatedwith .Thereproducingkernelsaredetermineduniquelybytheirsalientproperty: Z K n x;t P t d t = P x foranypolynomial P ofdegreelessthan n .Let P n denotethebestuniformapproximantof f bypolynomialsofdegreeatmost n )]TJ/F19 11.9552 Tf 12.343 0 Td [(1,witherror E n f .Theerrorof approximationof f byitspartialsumsisdeterminedby j S n )]TJ/F27 11.9552 Tf 11.294 0 Td [(f j .Since L n P k = P k j S n )]TJ/F27 11.9552 Tf 11.955 0 Td [(f j x j S n )]TJ/F27 11.9552 Tf 11.955 0 Td [(P n j x + j P n )]TJ/F27 11.9552 Tf 11.955 0 Td [(f j x n x +1 E n f ; Now, n x = Z j K n x;t j d t C Z j K n x;t j 2 d t 1 = 2 = C n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 j p k x j 2 1 = 2 = C n ;x 1 = 2 ; .1.4 where C isthecomplexplane.Thus,lowerboundsontheseriesofChristoelfunctionsimplyupperboundsontherateofconvergenceofaFourierseries.Infact,the ChristoelandLebesguefunctionssustainamoreintimaterelationshipthanthetrivialinequality,.1.4,suggests:Foraclassofmeasures, ,supportedontheinterval, [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1],P.Nevaiprovesin[14,p.12]that n ;x n x 2 = o as n !1 .The asymptoticbehaviorofthesequenceofChristoelfunctionshasprofoundimplications forapproximationtheory.Foranotherperhapstheoldestexample,letusconsider integralquadrature[4]. 1.2Quadrature Presumethatthecompactsupportof liesintheinterval[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1]sothatthezeros of p n x ,denoted x 1
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j 6 = k andequals1at x k .Anypolynomialofdegreeatmost n canbeprecisely interpolatedat f x k g n k =1 asfollows: p x = n X k =1 p x k l k x : Since f l k g isanorthogonalsetwithrespectto ,thisimpliesthefollowingquadrature formula: Z p x 2 d x = n X k =1 p x k 2 nk ; where nk = R l 2 k d aretheChristoel-Cotesnumbers.TheChristoel-Darboux identityisfundamentalformeasuressupportedontherealline: K n x;t = c n p n x p n +1 t )]TJ/F27 11.9552 Tf 11.956 0 Td [(p n t p n +1 x x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t ; .2.5 where c n isconstant.Itimpliesthat K n x j ;x k =0whenever j 6 = k .Therefore, l k x = n ;x k K n x;x k .Squaringbothsidesandintegratingagainst gives nk = n ;x k : TheasymptoticbehaviorofChristoelfunctionsisclearlycentraltothe convergenceofquadratureschemes. 1.3Universality Theasymptoticbehaviorofthereproducingkernels, K n ,playsasignicantrolein thetheoryofrandommatricesandstatisticalmechanics.Theso-calleduniversality" limitisespeciallyimportant:Forcertainclassesofmeasures, lim n !1 K n x + a=n;x + b=n K n x;x = sin )]TJ/F19 11.9552 Tf 5.48 -9.683 Td [( a )]TJ/F27 11.9552 Tf 11.956 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 a )]TJ/F27 11.9552 Tf 11.955 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 ; .3.6 uniformlyas a and b rangeoveracompactinterval.D.S.Lubinskyrecentlyadvanced thefrontierofuniversalitylimitsbymeansofaverysimplerelationshipbetween K n andtheChristoelfunctions, n .See[10].Fortwomeasures andtheir 4

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associatedChristoelfunctions n and n j K n x;y )]TJ/F27 11.9552 Tf 11.956 0 Td [(K n x;y j K n x;x n x n y 1 = 2 1 )]TJ/F27 11.9552 Tf 13.15 8.088 Td [( n x n x 1 = 2 : Universalityfollowsifthetranslatedlimitlim n !1 n n ;x + a=n canbecalculated. Lubinskydoesso,andproves.3.6foralargeclassofmeasures.Amongtheprimary contributionsofthisdissertationistheextensionofLubinsky'stechniquetoastill broaderclassofmeasures{atpresent,themostgeneralclass{Szeg}o'sclass[3].Our nalapplicationconcernstheasymptoticspacingofthezerosof p n 1.4Zero-Spacing Thespacingofthezeros, x nn
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1.5ABriefHistory Theasymptoticbehaviorof f n ;z g n diersdramaticallyaccordingtowhether z iscontainedinthesupportof ornot.If,forinstance, issupportedontheunit circleand j z j < 1,Szeg}oproved[11,p.434]that lim n !1 n ;z =1 )-222(j z j 2 j D ;z j ; where D ;z istheSzeg}ofunctionassociatedwith D ;z =exp 1 4 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( e i + z e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(z log 0 d ; j z j < 1 : Onthesupport,when j z j =1,thesituationismuchsimpler: n ;z f z g Whatistherateofthisconvergence?Szeg}oagainfoundthecorrectformula:for absolutelycontinuousmeasureswithsucientlysmooth,positiveweights, 0 lim n !1 n n ;e i = 0 : .5.9 Aswehaveseen,thisresulthasprofoundimplicationsforavarietyofsubjects.Consequently,manyhaveattemptedtoextendSzeg}o'scalculationtoever-broaderclasses ofmeasures. ThemostpreciseasymptoticformulasforthesequenceofChristoelfunctions weregivenbyFreudinhisdissertationandreiteratedinhistreatiseonorthogonal polynomials,[4,p.271].Let beanabsolutelycontinuousmeasuresupportedon [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]whoseweight, 0 ,isin Lip andsatises 0 x Q x 2 forsomepolynomial Q x .Heprovesthethefollowingrepresentation[4,p.254]: 1 n ;x = n 1 0 x p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 + O n 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( < 1 1 n ;x = n 1 0 x p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 + O log n =1 : 6

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Inparticular, lim n !1 n n ;x = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; .5.10 theanalogueofSzeg}o'sresultformeasuressupportedon[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1].Infact,thetwo resultsareequivalentbyvirtueofasimpleformulawewillciteinthenextchapter. AfterFreud,thenextsubstantialadvancementwasmadebyMateandNevaiin[12]. Szeg}o'sclassconsistsofmeasuressupportedontheunitcirclewhoseweightssatisfy Szeg}o'scondition, Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( log 0 d> : .5.11 MateandNevainearlyobtainedafullgeneralizationof.5.9fortheSzeg}oclass: theyproved,foralmostevery 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ],that e 0 liminf n !1 n n ;e i limsup n !1 n n ;e i 0 : .5.12 Theirresultconstitutesadramaticimprovement.Althoughtheydonotobtainthe precisevalueofthelimit.5.9,theirhypothesis|Szeg}o'scondition|isfarlessrestrictivethanFreud's:itmakesnoassumptionsonthesmoothnessoftheweight. Inparticular,.5.12appliestoeverymeasurewhoseweightisboundedbelowbya positiveconstant. Theholygrailwasnallyrecoveredin1991bythecircuitousandtechnicalargumentofMatewhichestablishes.5.9forSzeg}o'sclassinhislandmarkpaper authoredjointlywithPaulNevaiandVilmosTotik[11].NevaiusesMate'sresultto prove.5.10viathesimpleformulatowhichwehavealreadyalluded.Totikfurther showsthatSzeg}o'sconditionisunnecessarilyrestrictive:log 0 needonlybeintegrable overanopenintervalcontaining x .Underthiscondition,heproves.5.10forthe muchlargerclassofregularmeasures,studiedbyUllman[23].Thisclassisfar broaderthanSzeg}o's: isregularifitsChristoelfunctionssatisfytheverygeneral conditionliminf n !1 n ;x 1 =n 1.Sincetheirpaper,nofurthergeneralizations havebeenproved. Thisdissertationcomprisestwopapersbytheauthor[2,3]whichsubstantially 7

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improveupontheresultofMate[11]inthefollowingaspects:In[3],wecombine Mate'sapproachwiththatofD.Lubinskyin[10]toprovethemostgeneraluniversality limittodate.In[2],weadaptMate'stechniquetoobtain.5.9fortheclassof measuressupportedonsmoothcurvessatisfyingamodiedSzeg}ocondition.Thisis amongtherstcalculationoftheprecisevalueoflim n !1 n n ;z formeasureswith generalsupports.Totikwasthersttocomputethisvalueformeasuressupported ondisjoint,realintervals[21].Theonlyotherresultofthistypewasobtainedby GolinskiiforChebyshev-typeweightsoncirculararcs[6]. 8

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2Universality Inthischapter, denotesaniteBorelmeasuresupportedoneithertheunitcircle equivalentlyon )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ortheinterval[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]. 0 istheweightassociatedwithits absolutelycontinuouspart.If issupportedon[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1],theChristoelfunctionsare denedby n ;x =inf Z j P n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j 2 d; wheretheinmumistakenoverallcomplexpolynomialsofdegreeatmost n )]TJ/F19 11.9552 Tf 10.146 0 Td [(1which equalunityat x .Formeasuressupportedontheunitcircle,theintegralisevaluated withrespectto = 2 ,andtheChristoelfunctionsaredenotedby n ;z Let f p n g betheorthonormalpolynomialsassociatedwiththemeasure .For supportedontheinterval,[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1],theyaredeneduptoaconstantmultipleofunit modulusbytheconditions Z p n t t k d t =0and Z j p n j 2 d =1 : forall0 k
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Thereproducingkernelsaresonamedbecauseoftheirsalientfeature:Forallpolynomials, P ,withdegreeatmost n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1, Z P x K n x;t d x = P t : WewillinvestigatemeasureswhoseweightssatisfySzeg}o'sconditionlocally,that is, Z I log 0 d> ; forsomeinterval I .Ourresultsrequirethatthemeasuresalsobe regular inthesense ofUllman[23].Foracomprehensivetreatmentofthetheoryofregularmeasures, seethebookbyStahlandTotik[19].Regularityofameasure withcompact support K isequivalenttothefollowingcondition:Foreverysequence, f P n g 1 n =1 ; of polynomialswhosedegreesarenotgreaterthantheirindices, limsup n !1 k P n k K k P n k 1 =n 1 ; .0.2 where k P k 2 = Z j P j 2 d and k P k K =sup z 2 K j P z j : TheclassofregularmeasuresisfarlargerthanSzeg}o'sclass. D.S.Lubinsky,in[10],establishedthefollowinginequalityrelatingthereproducing kernelsoftwomeasures, ,totheirassociatedChristoelfunctions, and : j K n x;y )]TJ/F27 11.9552 Tf 11.956 0 Td [(K n x;y j K n x;x n x n y 1 = 2 1 )]TJ/F27 11.9552 Tf 13.15 8.088 Td [( n x n x 1 = 2 : HealsoprovesthefollowingasymptoticformulafortranslatedChristoelfunctions ofregularmeasureswhoseweightsarepositiveandcontinuousonsomeinterval, I : lim n !1 n n x + a=n = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; .0.3 uniformlyfor x 2 I and a inacompactsubsetof R .Withthesetwoformulae,he 10

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obtainsauniversalityresultfortheaforementionedclassofmeasures.Thegoalof thischapteristoproveLubinsky'sresultforthebroaderclassofregularmeasures whichsatisfySzeg}o'sconditionlocally.ThisisasubstantialrelaxationofLubinsky's hypotheses:itobviatestherequirementofcontinuityinfavorofalessrestrictivelocal Szeg}ocondition. Theorem2.0.1 Let bearegularBorelmeasureon [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] satisfyingSzeg}o'scondition, Z I log 0 t dt> onanopeninterval I [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] .Fix A> 0 .Then,foralmostevery x 2 I lim n !1 K n x + a=n;x + b=n K n x;x = sin )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( a )]TJ/F27 11.9552 Tf 11.956 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 a )]TJ/F27 11.9552 Tf 11.955 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 ; .0.4 uniformlyfor a;b 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] ThevehicleofthisextensionistheresultofMate,NevaiandTotik[11]whichestablishesthelimit.0.3with a =0forregularmeasuressatisfyingSzeg}o'scondition onaninterval I atalmostevery x 2 I .Wewillextendthetechniquesoftheseauthorstoobtain.0.3forthebroaderclassofregular,locallySzeg}omeasuresand thenmimicLubinsky'sproceduretoestablishuniversalityon I .Finally,weadapt thetechniqueofLevinandLubinsky[9]toprovearesultonthedistributionofthe zerosoforthogonalpolynomialsassociatedwithlocallySzeg}oweights. 2.1TranslatedAsymptotics Inthissectionweestablishtheasymptotics.0.3ofthetranslatedChristoelfunctionsformeasuresontheunitcirclewhichsatisfySzeg}o'scondition. Let d x = 0 x dx + d s x ,where s isthesingularpartof withrespectto Lebesguemeasureand 0 x dx isitsabsolutelycontinuouspart.Itisknownsee[17, Theorem8.6]that 0 t =lim 0 [ t;t + ] .1.5 11

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foralmostall t here,for < 0,dene [ t;t + ] = by [ t + ;t ] = j j .Recallthat t isaLebesguepointof 0 if lim 0 1 Z 0 j 0 t + u )]TJ/F27 11.9552 Tf 11.956 0 Td [( 0 t j du =0 : .1.6 Weusethefollowingterminology: t isa Lebesguepointof ifthelimitin.1.5 existsat t ,andwiththislimitas 0 t ,.1.6istrue.Thus,almostallpointsare Lebesguepointsof Theorem2.1.1 Let beameasureon )]TJ/F27 11.9552 Tf 9.299 0 Td [(; satisfyingSzeg}o'scondition.Fix A> 0 .Then,foralmostevery t 2 )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ,wehave lim n !1 n! n )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(;e i t + a=n = 0 t ; .1.7 uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] Furthermore, : 1 : 7 holdsatevery t whichisaLebesguepointof andforwhich e it isaLebesguepointoftheSzeg}ofunction see : 1 : 9 associatedwith TheupperlimitactuallyholdsforallniteBorelmeasures.Thisfollowsfromthe nextlemma,animprovementofLebesgue'sresultontheconvergenceofFejermeans see[17,p.244].Inwhatfollows, n ;z isthe n -thFejermeanofthemeasure givenby n ;z = Z F n z )]TJ/F27 11.9552 Tf 11.955 0 Td [(t d t withnormalizedkernels F n t = 1 2 n +1 sin 2 n +1 t= 2 sin 2 t= 2 : Lemma2.1.2 Let beanabsolutelycontinuousBorelmeasureon )]TJ/F27 11.9552 Tf 9.298 0 Td [(; suchthat 0 :=lim t & 0 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(t;t ] 2 t ; exists.ThenthetranslatedFejermeans n ;e ia=n 0 as n !1 uniformlyfor 12

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a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ] TheproofrequiresaresultconcerningHardy'smaximalfunctionwhichisdened{for ameasure supportedontherealline{by M x =sup t> 0 [ x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;x + t ] 2 t : Lemma2.1.3 Let f beaneven,positivefunctionon [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ] ,decreasingawayfrom 0 .Then,foranymeasure supportedonaninterval I =[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(t;t ] [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ] ,wehave Z I fd M Z I f t dt: Proof. Let I = I 1 I 2 I n beanestedsequenceofsymmetricintervalsand choosepositivenumbers a 1 ;:::;a n ,suchthat f t s t := P k a k I k t .Then Z fd X k a k I k M X k a k j I k j = M Z s t dt; where jj denotesLebesguemeasure.Since f decreasesawayfrom0andiseven, thereisasequenceofsimplefunctions s n oftheform s n = P k a k I k whichdominate f andforwhich R s n t dt R f t dt .Thisestablishesthelemma. Proof. Lemma2.1.2Withoutlossofgeneralityassume 0 =0subtracta constantifnecessary.Webeginwiththeclaimthatforsomeconstant C> 0, F n t min n +1 ; 2 n +1 t 2 C n +1 1+ n +1 2 t 2 .1.8 forevery t 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ]and n 0.Toseetherstinequality,observethat sin t 2 1 j t j forall t 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ].Thus, F n t 1 2 n +1 2 t 2 : Butthekernel F n istheaverageoftherst n +1Dirichletkernels,soitsmaximumis achievedat0andisequalto n +1 = 2 .Thisprovestherstinequality.Thesecond 13

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inequalityistrueforany C> 1+ = 2.Indeed, n +1 < 2 n +1 t 2 C n +1 1+ n +1 2 t 2 > C n +1 1+ = 2 >n +1 : Ontheotherhand, x 2 + x + x 1 1+ x = + x ; andso 2 n +1 t 2 n +1 n +1 2 t 2 2 C n +1 1+ n +1 2 t 2 C= + n +1 t 2 1 n +1 t 2 + = 2 + = 2 n +1 t 2 : Thisestablishes.1.8. Now,choose > 0andlet I 0 =[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(a 0 ;a 0 ]beanintervalcenteredat0suchthat I < j I j foreverysymmetricinterval I I 0 .Dene 0 = I 0 and 1 = )]TJ/F27 11.9552 Tf 10.061 0 Td [( 0 .We showthat n 0 ;e ia=n and n 1 ;e ia=n convergetozerouniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ]. Since = 0 + 1 ,thiswillestablishthelemma.Using.1.8,wend n 1 ;e ia=n = Z [ )]TJ/F28 7.9701 Tf 6.586 0 Td [(; ] n I 0 F n t )]TJ/F27 11.9552 Tf 11.956 0 Td [(a=n d t Z j t j >a 0 f a n t d t ; wherethefunctions f a n t = C n +1 1+ n +1 2 t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n 2 tendtozerouniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ]and j t j >a 0 .Sodotheirintegrals,which establishestheconvergencefor 1 Tohandletheintegralwith 0 ,dene L n t :=sup a 2 [ )]TJ/F28 7.9701 Tf 6.586 0 Td [(A;A ] f a n t = 8 > > < > > : C n +1 ; if j t j A=n C n +1 1+ n +1 2 j t j)]TJ/F28 7.9701 Tf 8.938 0 Td [(A=n 2 ; if j t j >A=n: 14

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Thefunctions L n areevenanddecreasingawayfrom0,sowemayapplyLemma2.1.3 togetherwiththeestimate F n t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n L n t toobtain n 0 ;e ia=n = Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( F n t )]TJ/F27 11.9552 Tf 11.956 0 Td [(a=n d 0 t Z I 0 L n t d t j I 0 j Z I 0 L n t dt; forall a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ].But, Z I 0 L n t dt Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( L n t dt = AC n +1 n +2 Z A=n C n +1 1+ n +1 2 t )]TJ/F27 11.9552 Tf 11.955 0 Td [(A=n 2 dt 2 AC +2 Z 1 0 C 1+ u 2 du whichisniteandindependentof n .Since isarbitrary,thiscompletestheproof. Lemma2.1.4 Let beaniteBorelmeasureon [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ] .Then,ateveryLebesgue point t 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ] of 0 limsup n !1 n! n )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(;e i t )]TJ/F28 7.9701 Tf 6.586 0 Td [(a=n 0 t ; uniformlyforall a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ] Proof. Withoutlossofgenerality,wemayassumethat isabsolutelycontinuous. FixaLebesguepointof 0 t 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ],anddenethepolynomial P by P = 1 n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X j =0 e )]TJ/F28 7.9701 Tf 6.586 0 Td [(ij t )]TJ/F28 7.9701 Tf 6.586 0 Td [(a=n j : P e i t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n =1and P e i 2 = sin n t )]TJ/F28 7.9701 Tf 6.586 0 Td [(a=n )]TJ/F28 7.9701 Tf 6.586 0 Td [( 2 n sin t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n )]TJ/F28 7.9701 Tf 6.587 0 Td [( 2 2 = 2 n F n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n )]TJ/F27 11.9552 Tf 11.955 0 Td [( ; 15

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sothat n! n )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(;e i t )]TJ/F28 7.9701 Tf 6.586 0 Td [(a=n n 2 Z P )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(e i 2 d t = n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(;e i t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n ByLemma2.1.2,therighthandsideconvergesto 0 t uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ], whichcompletestheproof. TheproofofthelowerboundreliesheavilyontheSzeg}ofunctionassociatedwith themeasure : D z = D z =exp 1 4 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( e i + z e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(z log 0 d ; j z j < 1.1.9 FormeasuressatisfyingSzeg}o'scondition,thisfunctionisinHardy'sclass, H 2 seee.g. [17,242-244],where D 2 iscalledtheouterfunctionassociatedwith 0 .Thefollowing propertieswillbeimplicitlyinvokedthroughouttheproof: D z hasnontangential limit D e it atalmosteverypoint z = e it ,whichsatises j D e i j 2 = 0 atalmost every 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ].Inparticular,thenontangentiallimitexistsatevery z = e it which isaLebesguepointfor D e i seeFatou'stheorem[7,p.34]andapplyittothe complexvaluedharmonicfunction D .Finally, Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( D e i d =lim r 1 )]TJ/F32 11.9552 Tf 8.246 22.084 Td [(Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( D re i d: Proof. Theorem2.1.1 Weprovethatforalmostevery e it 2 T liminf n !1 n! n ;e i t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n 0 t .1.10 uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ].ThistogetherwithLemma2.1.4provesTheorem2.1.1. Withoutlossofgeneralitywemayassumethat isabsolutelycontinuous,sinceif not,themonotonicityoftheChristoelfunctionsimpliesthat n ;z n 0 ;z Inthiscase,.1.10onlyincreases.Weshallshowthat.1.10holdsateverypoint t whichisaLebesguepointof 0 andforwhich e it isaLebesguepointoftheSzeg}o 16

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function D e i .Thus,let t besuchapoint.Wemayassumethat e it =1andhence that lim 0 1 Z 0 j 0 u )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 j du =0 and lim 0 1 Z 0 j D e iu )]TJ/F27 11.9552 Tf 11.955 0 Td [(D j du =0 : Theseimplythatthereisaset S with0asadensitypointsothatthelimitat0of 0 u along S is 0 ,whilethelimitof D e iu along S is D .Thiscombinedwith j D e iu j 2 = 0 u a.e.impliesthat j D j 2 = 0 Sincewewanttoprovethelowerestimate.1.10,wemayalsoassumethat fortheparticular n 2 N appearingintheproofand a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ]theinequality n! n ;e )]TJ/F28 7.9701 Tf 6.586 0 Td [(ia=n j D j 2 = 0 holds.For a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ]and n 2 N ,let q = q a;n = e )]TJ/F28 7.9701 Tf 6.587 0 Td [(ia=n andchoosepolynomials, P = P a;n ,ofdegreeatmost n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1forwhich n ;q = 1 2 j P q j 2 Z P )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(e i 2 d : Nowxasmall > 0and > 4 = 2 withalso > 2 A anddene K 1 =[ )]TJ/F27 11.9552 Tf 9.298 0 Td [(=n;=n ] and K 2 =[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ] n K 1 Weclaimthatforsucientlylarge n j P e i j 3 j P q j forall 2 K 1 and a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ].If j j =1,and =1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 =n ,thenforany n and a j P 2 D 2 j = 1 2 i I j z j =1 P 2 z D 2 z z )]TJ/F27 11.9552 Tf 11.955 0 Td [( dz 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( j PD j 2 e i 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( d = 1 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( P )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e i 2 d = 1 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( j P q j 2 n ;q j P q j 2 j D j 2 : Itisknownthatthezerosof P lieontheunitcircle, T ,soitiselementarysee[11, page438]that j P j 1+ 2 n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j P j = 1 )]TJ/F19 11.9552 Tf 16.645 8.088 Td [(1 2 n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j P j 1 2 j P j .1.11 17

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Thus, j P 2 D 2 j 4 j P q j 2 j D j 2 forall n 2 N a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ],and j j =1.Now1= e i 0 isaLebesguepointof D e i ,so D z hasanontangentiallimit D at1.Hence,forlarge n ,andarg 2 K 1 j D j 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( j D j 2 and,therefore, j P j 2 p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( j P q j 3 j P q j ; .1.12 whichprovestheclaim. Nowlet r =1+ =n .Weshowthat j I 1 =D )]TJETq1 0 0 1 305.42 469.617 cm[]0 d 0 J 0.478 w 0 0 m 40.75 0 l SQBT/F19 11.9552 Tf 306.846 460.008 Td [(~ I 1 =D j < 4 j P q j .1.13 where I 1 = 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 D e i d ~ I 1 = 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 D e i d : Tothisend,dene I 1 j = 1 2 Z K j P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 D e i d ~ I 1 j = 1 2 Z K j P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 D e i d 18

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Weestablish.1.13byanalyzingthefollowingdecomposition: I 1 =D )]TJETq1 0 0 1 175.183 685.279 cm[]0 d 0 J 0.478 w 0 0 m 40.75 0 l SQBT/F19 11.9552 Tf 176.608 675.671 Td [(~ I 1 =D = I 11 D )]TJ/F19 11.9552 Tf 16.685 8.088 Td [(1 2 Z K 1 P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 d + I 12 D + )]TJ/F19 11.9552 Tf 17.356 11.11 Td [(~ I 11 D + 1 2 Z K 1 P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 d )]TJETq1 0 0 1 463.605 644.207 cm[]0 d 0 J 0.478 w 0 0 m 27.393 0 l SQBT/F19 11.9552 Tf 471.663 634.599 Td [(~ I 12 D : .1.14 Toestablishanupperboundforthersttermontherighthandside,observethat 1 2 Z K 1 P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(D e i )]TJ/F27 11.9552 Tf 11.956 0 Td [(D d max 2 K 1 P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 1 2 Z K 1 D e i )]TJ/F27 11.9552 Tf 11.956 0 Td [(D d .1.15 By.1.12,themaximumis max 2 K 1 P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 3 j P q j r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n j r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 3 j P q j n ; andforlarge n ,bytheLebesguepointproperty, Z K 1 D e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(D d< 2 6 2 n = 2 3 n ; sothat.1.15is j P q j .Butthen, I 11 D )]TJ/F19 11.9552 Tf 16.685 8.088 Td [(1 2 Z K 1 P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 d P q D .1.16 Ananalogousargumentyields )]TJ/F19 11.9552 Tf 17.355 11.11 Td [(~ I 11 D + 1 2 Z K 1 P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(in r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 d P q D ; .1.17 forthethirdtermof.1.14.Forthesecondtermof.1.14theCauchyinequality gives j I 12 j 2 1 2 Z K 2 j P e i j 2 j D e i j 2 d 1 2 Z K 2 r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i q )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 2 d: 19

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Therstintegralis 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( j P e i j 2 j D e i j 2 d = 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( j P e i j 2 d = n q j P q j 2 Thesecondintegralis 1 2 Z K 2 r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n +1 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + a=n )]TJ/F27 11.9552 Tf 11.955 0 Td [(r 2 d 1 2 Z K 2 1 j e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + a=n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 2 d = 1 2 Z K 2 1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2cos + a=n d = 1 2 Z K 2 1 sin + a=n = 2 2 d 1 2 Z K 2 2 + a=n )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 d 1 Z 1 =n 2 2 d = n ; since2 j + a=n jj j on K 2 .Thus, j I 12 j 2 n n q j P q j 2 1 j D j 2 j P q j 2 2 j D j 2 j P q j 2 ; sothat j I 12 j j P q jj D j : .1.18 Ananalogousargumentestablishesthat j ~ I 12 j j P q jj D j : .1.19 Equations.1.16,.1.17,.1.18,and.1.19prove.1.13. If P = P n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 k =0 c k k ,let ~ P P n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 k =0 c k n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(k .For = e i ~ P = P n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ,so ~ I 1 = 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( ~ P e i r )]TJ/F28 7.9701 Tf 6.586 0 Td [(n q n e i r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 e i q )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 D e i d = 1 2 i q r n I j j =1 ~ P D r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 d But r> 1,so F = ~ P D = r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F19 11.9552 Tf 11.365 0 Td [(1isholomorphicin= fj z j < 1 g andhas 20

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nosingularitiesthere.Also, D 2 H 2 ,hencesois F ,andtherefore, I j j =1 F d =lim 1 )]TJ/F32 11.9552 Tf 8.246 22.084 Td [(I j j =1 F d =0 ; which,togetherwith.1.13,impliesthat j I 1 j < 4 j P q jj D j : Finally,let H n z = z )]TJ/F28 7.9701 Tf 6.587 0 Td [(n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 z )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 = n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 z )]TJ/F28 7.9701 Tf 6.587 0 Td [(k Then, 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( P e i H n rq )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 e i D e i d = 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( P e i r )]TJ/F28 7.9701 Tf 6.587 0 Td [(n q n e )]TJ/F28 7.9701 Tf 6.587 0 Td [(in )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 qe )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 D e i d = I 1 )]TJ/F19 11.9552 Tf 16.686 8.087 Td [(1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( P e i D e i e i r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F27 11.9552 Tf 11.956 0 Td [(e i d = I 1 )]TJ/F19 11.9552 Tf 18.682 8.087 Td [(1 2 i I j j =1 P D r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F27 11.9552 Tf 11.955 0 Td [( d = I 1 + P r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q D r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q ; sothatforlarge n P r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q D r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F19 11.9552 Tf 16.685 8.088 Td [(1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( P e i H n rq )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 e i D e i d 4 j P q D j But, 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( P e i H n rq )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 e i D e i d 2 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( j H n rq )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 e i j 2 d 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( j P e i D e i j 2 d = n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 1+ n )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( j P e i j 2 d n j P q j 2 n q 21

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Thus, j P r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q D r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q j)]TJ/F19 11.9552 Tf 17.932 0 Td [(4 j P q jj D jj P q j p n! n q But,asin.1.11,wehave j P r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q j 1+1 =r 2 n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j P q j = 2+ =n 2+ =n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j P q j = 1 )]TJ/F27 11.9552 Tf 30.765 8.088 Td [(=n 2+ =n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j P q j )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(=n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j P q j e )]TJ/F28 7.9701 Tf 6.586 0 Td [( j P q j Also,as n !1 r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q 1non-tangentially,sothat D r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q D ,andtherefore e )]TJ/F28 7.9701 Tf 6.586 0 Td [( )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 j D j liminf n !1 p n! n q Thiscompletestheproofof.1.10since > 0isarbitrary. 2.2Measureson [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] On[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1]theSzeg}oclassconsistsofallniteBorelmeasures withsupporton [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]forwhich Z 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 log 0 x p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 dx> ; .2.20 Formeasuresinthisclass,theSzeg}ofunctionisdenedon C n [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]by ~ D z = ~ D z =exp p z 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 1 2 Z 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 log 0 x z )]TJ/F27 11.9552 Tf 11.955 0 Td [(x dx p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 .2.21 withthebranchofthesquarerootthatispositiveforpositive z .Thishasnontangentiallimit ~ D x atalmostevery x 2 [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]with j ~ D x j 2 = 0 x Theorem2.1.1translatesappropriatelyformeasuresontheinterval[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1]. Theorem2.2.1 Let beameasureon [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] satisfyingSzeg}o'scondition.Then, uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] lim n !1 n n ;x + a=n = 0 x p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 .2.22 22

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foralmostevery x 2 [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] .Moreover, : 2 : 22 holdsatevery x 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1 whichisa Lebesguepointof andfor ~ D Theproofreliesonorthogonalpolynomials.Denetheintegraloperators, G n ,by G n f;x = n ;x Z f t K 2 n ; x;t d t ; where K n arethereproducingkernelsdenedintheintroduction.Thenextresults arefoundin[13]aswellas[15,p.230,Corollary4.3.1]. Lemma2.2.2 Let f becontinuouson [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] andlet beameasureon [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] which satisesSzeg}o'scondition.Then, lim n !1 sup x 2 [ )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; 1] j G n f;x )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x j =0 : Lemma2.2.3 Let beameasureon [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] satisfyingSzeg}o'scondition.Thenfor anyxed m 1 wehave lim n !1 sup x 2 [ )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; 1] n + m ;x n ;x )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 =0 : Weproveanextensionofacorollarygiveninthesamepaper. Corollary2.2.4 Let g beanonnegativefunctionon [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] andassumethatthere existsapolynomial P m forwhich P m g and P m g )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 arecontinuouson [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] .Let d g x = g x d x ,where isameasureon [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] satisfyingSzeg}o'scondition. Then lim n !1 n g ;x + a=n n ;x + a=n = g x uniformlyineverycompactsubsetof )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1 devoidofzerosof P m anduniformlyfor all a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] 23

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Proof. G n ; gP 2 m ;x = n ;x Z P 2 m t g t K 2 n ; x;t d t = n ;x Z P 2 m t K 2 n ; x;t d g t n ;x n + m g ;x P 2 m x K 2 n ; x;x = n + m g ;x n ;x P 2 m x : Thenalequalityfollowsfrom.0.1.Thus,uniformlyin a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] limsup n !1 n + m g ;x + a=n n ;x + a=n lim n !1 G n ; gP 2 m ;x + a=n P 2 m x + a=n = g x .2.23 uniformlyoncompactsubsetsof )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1devoidofzerosof P m .Also, G n g ;P 2 m =g;x = n g ;x Z P 2 m t K 2 n g ; x;t d t n g ;x m + n ;x P 2 m x K 2 n g ; x;x = n + m ;x n g ;x P 2 m x So,asabove, limsup n !1 n + m ;x + a=n n g ;x + a=n lim n !1 G n g ;P 2 m =g;x + a=n P 2 m x + a=n = 1 g x .2.24 locallyuniformlyin )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1awayfromzerosof P m anduniformlyin a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ], since P m =g 2 L 1 and g alsosatisesSzeg}o'scondition.Theresultnowfollowsfrom Lemma2.2.3andinequalities.2.23and.2.24. TheproofofTheorem2.2.1reliesonanequationwhichrelatestheChristoel functionofameasure, ,supportedontheinterval[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]toits`projection'ontothe unitcircle.Dened by E := f cos : 2 E g for E [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; 0or E [0 ; .Notethat 0 t = 0 cos t j sin t j .See[11,Lemma6,p. 446]foraproofofthefollowing: 24

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Lemma2.2.5 GivenanarbitrarypositiveniteBorelmeasure on [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] ,forevery integer n> 1 andevery t 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; 1 2 n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;e it = n ; cos t + sin 2 t n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 g ; cos t ; where g x =1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 Withthesepreliminaries,wecanproceedwiththeproofofTheorem2.2.1. Proof. Theorem2.2.1As n !1 cos t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n =cos t + a n sin t + O 1 n 2 =cos t + a n O : FromLemma2.2.3andCorollary2.2.4,itfollowsthat lim n !1 n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 g ; cos t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n n ; cos t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n =sin 2 t uniformlyoncompactsubsetsof )]TJ/F27 11.9552 Tf 9.299 0 Td [(; nf 0 g and a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ].Consequently,Lemma 2.2.5gives lim n !1 2 n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;e i t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n n ; cos t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a=n = 1 2 ; .2.25 uniformlyoncompactsubsetsof )]TJ/F27 11.9552 Tf 9.299 0 Td [(; nf 0 g and a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ].Wewritecos t )]TJ/F27 11.9552 Tf -424.076 -20.922 Td [(a=n =cos t + b=n ,andnotethat,while a runsthrough[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ],the b = b a;t covers aninterval[ )]TJ/F27 11.9552 Tf 9.298 0 Td [(B A;t ;B A;t ]dependingon t 2 )]TJ/F27 11.9552 Tf 9.298 0 Td [(; nf 0 g andon A ,andhereforany t 2 )]TJ/F27 11.9552 Tf 9.299 0 Td [(; nf 0 g andany B> 0thereisan A suchthat[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(B;B ] [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(B A;t ;B A;t ].The sameistrueuniformlyif t runsthroughacompactsubsetof )]TJ/F27 11.9552 Tf 9.299 0 Td [(; nf 0 g .So,since theconvergencein.2.25isuniformoveranyinterval[ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ],weget lim n !1 2 n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;e i t )]TJ/F28 7.9701 Tf 6.587 0 Td [(a=n n ; cos t + b=n = 1 2 ; uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ].Therefore,byTheorem2.1.1, lim n !1 n n ; cos t + b=n = 0 t ; 25

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foralmostevery t uniformlyin b 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(B;B ].Substitutingcos t with x gives lim n !1 n n ;x + b=n = p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 0 x ; foralmostevery x uniformlyin b 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(B;B ],whichcompletestheproofoftherst assertionofTheorem2.2.1. Toprovethelaststatement,weonlyneedtoobservethatundertheconformal map w = z )]TJ 12.234 9.834 Td [(p z 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1,thecomplementof[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]ismappedintotheunitdisk,and theSzeg}ofunction ~ D ismappedintotheSzeg}ofunction D see.1.9associated with .Therefore, x =cos t isaLebesguepointof andfor ~ D u preciselywhen t isaLebesguepointof and e it isaLebesguepointof D e iu .Takingtheseinto account,thelaststatementfollowsfromTheorem2.1.1. 2.3AsymptoticsforRegularMeasures TheassumptionsofTheorem2.2.1areunnecessarilyrestrictive.Regularmeasures denedintheintroductionby.0.2thatsatisfySzeg}o'sconditionlocally{on I ,an interval{generateChristoelfunctionsthatexhibittheasymptoticbehaviorof.0.3 when x 2 I .SinceweshallnowworkwithalocalSzeg}ocondition,werequirealocal Szeg}ofunction.Thus,letussupposethat isaniteBorelmeasureon[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1],and onsomeopeninterval I [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1]itsatisesSzeg}o'scondition,i.e. Z I log 0 t dt> : Wedene D z = D z =exp i 2 Z I log 0 x z )]TJ/F27 11.9552 Tf 11.955 0 Td [(x dx : .3.26 D z hasanontangentiallimitfromtheupperhalfplane D x atalmostevery x 2 I and j D x j 2 = 0 x a.e.seeLemma2.3.2below. Theorem2.3.1 Let bearegularmeasureon [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] andlet I beanopeninterval 26

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in [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] suchthat Z I log 0 d> : Then,foralmostevery x 2 I andforevery A> 0 lim n !1 n n ;x + a=n = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; .3.27 uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] Moreover, : 3 : 27 holdsatevery x 2 I whichisaLebesguepointof 0 andof D Proof. For > 0,let d x = d x + [ )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; 1] x dx ,where isthecharacteristic function.ThismeasureclearlysatisesSzeg}o'sconditiongloballyandtherefore,by Theorem2.2.1, limsup n !1 n n ;x + a=n = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x + : But < forevery andso,byvirtueofthemonotonicityoftheChristoelfunctions withrespecttomeasures, limsup n !1 n n ;x + a=n p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; foralmostevery x 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1anduniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ]. Toprovethelowerbound, liminf n !1 n n ;x + a=n p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; .3.28 uniformlyfor a 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ]andalmostevery x 2 I ,let d x = d x + [ )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; 1] n I x dx: InLemma2.3.2wewillprovethat x 2 I isaLebesguepointof D ifandonlyitis aLebesguepointoftheSzeg}ofunction ~ D associatedin.2.21with .Therefore, itisenoughtoshow.3.28atevery x 2 I whichisaLebesguepointof 0 andof 27

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~ D .Let x besuchapoint.Assumetothecontrarythattherearesequences, N N f a n 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ]: n 2Ng ,andarealnumber r< p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 forwhich n n ;x + a n =n 0,denethepolynomials Q n t = P n t 1 )]TJ/F32 11.9552 Tf 11.955 16.857 Td [( x + a n =n )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 4 2 [ n ] : Here[ y ]istheintegralpartof y .Evidently, Q n x + a n =n =1and j Q n t jj P n t j t 2 [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1].Furthermore,since a n =n 0as n !1 ,thereisa < 1suchthat Q n t P n t = 1 )]TJ/F32 11.9552 Tf 11.955 16.857 Td [( x + a n =n )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 4 2 [ n ] < 2 n n !1 .3.31 on[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] n I .Now,by.0.2,everyregularmeasure on[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1]hastheproperty thatforany s> 1andanysequenceofpolynomials, R n max t 2 [ )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; 1] j R n t j 2 1andsucientlylarge n 2N ,we have max t 2 [ )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; 1] j P n t j 2
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which,togetherwith.3.31andwiththeassignment s =1 = ,impliesthat j Q n t j < n on[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] n I forsucientlylarge n 2N .This,togetherwith.3.29and.3.30 impliesthat Z j Q n t j 2 d t r n 0 x + n Z [ )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; 1] n I dt = c n + 0 x + o =n ; where c = r + .Sincethisholdsforarbitrary ,wecanxitsvaluesothat c< p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 .Inthiscase [ n + ] Z j Q n t j 2 d t c 0 x + o for n 2N ; whichimpliesthat liminf n !1 n ;x + a n =n < p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 x ; since Q n x + a=n =1and Q n hasdegree[ n + ].ThiscontradictsTheorem2.2.1 at x andthiscontradictionprovestheclaim,pendingtheproofofthenextlemma. Lemma2.3.2 With d x = d x + [ )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; 1] n I x dx thenontangentiallimit fromthe upperhalfplane atan x 2 I existsfor D z ifandonlyifitexistsfor ~ D z Furthermore, x isaLebesguepointof D u preciselywhenitisaLebesguepointof ~ D u Proof. Supposerstthatthenontangentiallimitof D z existsat x 2 I .Consider thefunction ~ D z =exp p z 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 1 2 Z I log 0 x z )]TJ/F27 11.9552 Tf 11.955 0 Td [(x dx p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 : .3.32 Sincethisdierson I R from ~ D byananalyticmultiplicativefactor,itisenough toprovetheexistenceofthenontangentiallimitfor ~ D at x .But ~ D z =D z =exp i 2 Z I log 0 t h z;t dt .3.33 29

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with h z;t = 1 z )]TJ/F27 11.9552 Tf 11.955 0 Td [(x p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z 2 p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(t 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 = )]TJ/F27 11.9552 Tf 69.899 8.088 Td [(t + z p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 2 p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(z 2 + p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 2 ; whichisananalyticfunctionin z on I ,sothenontangentiallimit ~ D u =D u =exp i 2 Z I log 0 t h u;t dt .3.34 of.3.33certainlyexistsatany u 2 I .Thisshowsthat,indeed,thenontangential limitof ~ D = D ~ D =D alsoexistsat x Itisasimpleexercisetoshowthatif f isanonzero C 1 -function,then x isa Lebesguepointof D u ifandonlyifitisaLebesguepointof f u D u .With f u = ~ D u =D u thisisthesameas x beingaLebesguepointof ~ D u .Applying thesameargumentoncemorewith f u = ~ D u = ~ D u wendthat x isaLebesgue pointof D u ifandonlyifitisaLebesguepointof ~ D u .Theproofoftheconverse implicationi.e.goingfrom ~ D u to D u isverysimilar. Sincetherealpartof i= z )]TJ/F27 11.9552 Tf 12.144 0 Td [(t for z = x + iy isthePoissonkernel y= x )]TJ/F27 11.9552 Tf 12.145 0 Td [(t 2 + y 2 oftheupperhalfplane,itisastandardexercisetoshowthat j D z j 2 tends nontangentiallyto 0 x ateveryLebesguepointof 2.4Universality WewillnowapplyLubinsky'stechnique[10]toprovetheuniversalityresult,Theorem2.0.1.Theprooffollowsdirectlyfromthefollowing: Lemma2.4.1 Let and satisfytheconditionsofthehypothesisinTheorem2.0.1 andassumefurtherthat 0 x 0 = 0 x 0 > 0 forsome x 0 2 I whichisaLebesgue pointof , D and D see : 3 : 26 .Then lim n !1 1 n j K n )]TJ/F27 11.9552 Tf 11.955 0 Td [(K n j x 0 + a=n;x 0 + b=n =0 ; uniformlyfor a;b 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] ,where K n and K n arethereproducingkernelsassociated respectivelywith and 30

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Proof. Firstassumethat on[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1].Itwasprovenin[10,.5]that j K n x;y )]TJ/F27 11.9552 Tf 11.956 0 Td [(K n x;y j K n x;x s n x n y 1 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [( n x n x : .4.35 Here,the 'saretheassociatedChristoelfunctions.ByTheorem2.3.1 lim n !1 n n x 0 + a=n = q 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 0 x 0 .4.36 and lim n !1 n n x 0 + a=n = q 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 0 0 x 0 ; .4.37 uniformlyforall a 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ].Nowreplace x with x 0 + a=n and y with x 0 + b=n in .4.35.Then,since 0 x 0 = 0 x 0 ,itfollowsthat lim n !1 n x 0 + a=n j K n )]TJ/F27 11.9552 Tf 11.955 0 Td [(K n j x 0 + a=n;x 0 + b=n =0 ; uniformlyfor a;b 2 [ )]TJ/F27 11.9552 Tf 9.299 0 Td [(A;A ],whichimplies,againbecauseof.4.36,that lim n !1 1 n j K n )]TJ/F27 11.9552 Tf 11.955 0 Td [(K n j x 0 + a=n;x 0 + b=n =0uniformlyfor a;b 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(A;A ] : .4.38 Forarbitrary and satisfyingtheconditionsofthelemma,denethemeasure d x =maxdist x;I ; 0 x ; 0 x dx + d s x + d s x ; where 0 and s denote,respectively,theabsolutelycontinuousandsingularcomponentsofthemeasure .Clearly, ; so satisesSzeg}o'sconditionlocallyon I andisaregularmeasureon[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1].Hence.4.38holdsforthepairs ; and ; and,consequently,for ; .Thiscompletestheproof. 31

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Proof. Theorem2.0.1Let x beaLebesguepointof andof D see.3.26and assume 0 x > 0.Dene d u = 0 x du u 2 [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1].ByLemma2.4.1 lim n !1 1 n j K n )]TJ/F27 11.9552 Tf 11.955 0 Td [(K n j x + a=n;x + b=n =0 : .4.39 ApplyingLubinsky'soriginaltheorem[10,Theorem1.1]to K n wendthat,as n !1 K n x + = 0 x K n x;x ;x + = 0 x K n x;x K n x;x sin )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 11.956 0 Td [( ; .4.40 uniformlyforanyxed B and ; 2 [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(B;B ].Now,choose = n and = n so that = 0 x K n x;x = a=n and = 0 x K n x;x = b=n .Then,becauseof.4.37, as n !1 a= p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 and b= p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 ,hencethestatementinTheorem 2.0.1followsfrom.4.40and.4.39seealso.4.37. 2.5ZeroDistributionofOrthogonalPolynomials Finally,weapplythetechniquesofLevinandLubinskytoextendtheirTheorem1.1 in[9].TheapplicationofuniversalitytostudyzerospacingisalsofoundinFreud's text,[4]aswellasin[22].Inwhatfollows, x kn denotesthek-thzerooftheorthogonal polynomial p n associatedwithagivenmeasure ,denedontheinterval[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1].Let thezerosbeorderedaccordingto x nn 0 .Ifforsomesequence k = k n j x kn )]TJ/F27 11.9552 Tf 11.956 0 Td [(x j = O 1 n ; then lim n !1 x kn )]TJ/F27 11.9552 Tf 11.955 0 Td [(x k +1 ;n n p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 =1 : 32

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Proof. Let l kn betheLagrangeinterpolationpolynomialassociatedwiththepoint x kn whichvanishesateveryotherzeroof p n andsatises l kn x kn =1. l kn hasthe representation l kn z = K n x kn ;z K n x kn ;x kn : Thereisaboundedsequence, a n suchthat x kn = x + a n n : Since 0 x > 0,Theorem2.3.1impliesthat K n x kn ;x kn =K n x;x 1as n !1 sothatTheorem2.0.1appliedtotheLagrangepolynomialsgives l kn x + b n = sin a n )]TJ/F27 11.9552 Tf 11.955 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 a n )]TJ/F27 11.9552 Tf 11.955 0 Td [(b = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 + o ; .5.42 uniformlyforbounded b .Thersttermontherighthandsidetakenasafunctionof b ,changessignwhen a n )]TJ/F27 11.9552 Tf 11.53 0 Td [(b = )]TJ/F27 11.9552 Tf 9.298 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 andtherefore,by.5.41,thezero x k +1 ;n hastherepresentation, x k +1 ;n = x kn + b n n ; .5.43 forsomeboundedsequence b n < 0with liminf n !1 b n )]TJ/F27 11.9552 Tf 21.917 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 : So,by.5.42, 0= l kn x k +1 ;n = sin b n = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 b n = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 + o : .5.44 Weclaimthatlim n !1 b n = )]TJ/F27 11.9552 Tf 9.298 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 .Tothisendchooseanysubsequenceof f b n g withlimitpoint b .Equation.5.44gives,uponpassingthroughthissubsequence, sin b= p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 b= p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 =0 : 33

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Since )]TJ/F27 11.9552 Tf 9.298 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 b 0,wemusthave b = )]TJ/F27 11.9552 Tf 9.298 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 ,whichprovestheclaim. Thistogetherwithequation.5.43gives x k +1 ;n )]TJ/F27 11.9552 Tf 11.955 0 Td [(x kn n = b n !)]TJ/F27 11.9552 Tf 24.574 0 Td [( p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 ; as n !1 ,whichcompletestheproof. 34

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3ChristoffelFunctionAsymptoticsonGeneralCurves Wenowturnourdiscussiontomeasureswithgeneralsupportsintheplane.How doestheasymptoticbehaviorofthesequenceofChristoelfunctionsdependsonthe geometryofthesupport,)-277(:= supp ,whenthepointofevaluation, z 2 ?Itiseasy tosee,forinstance,that n ;z f z g as n !1 if)-317(issucientlyregulare.g. smoothand z liesontheouterboundaryof.Whatistherateofthisconvergence forgeneralmeasures,andhowisthisratedeterminedby?Formeasuressupported onacircleoraunionofintervals,wealreadyknowtheanswer: lim n !1 n n ;e i = 1 2 0 e i orlim n !1 n n ;x = p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(x 2 0 x .0.1 almosteverywhereonthesupports.Theseequationshintattheanswertoour question,buttounderstandthehintwerequiresomeelementarypotentialtheory. 3.1PotentialTheory Everymeasure, ,withcompactsupport,,hasassociatedlogarithmicenergydened asfollows: I := ZZ log 1 j z )]TJ/F27 11.9552 Tf 11.955 0 Td [(w j d z d w : If)-348(issucientlydense,then I >> 0forallprobabilitymeasures supported on.Thefundamentaltheoremofpotentialtheoryguaranteesaunique ,denoted )]TJ/F19 11.9552 Tf 5.787 1.793 Td [(,whoseenergyisleastamongallsuchmeasures.Itisknownasthelogarithmic equilibriummeasure,supportedontheouterboundaryof)-333(anddeterminedentirely bythegeometryofthisboundary.Letbetheunboundedcomponentof C n )]TJ -215.227 -35.616 Td [(35

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and g z; 1 ,itsGreen'sfunctionwithpoleatinnity.If)-341(issmooththenwemay exploitthefollowingusefulrepresentationofequilibriummeasure: d )]TJ/F19 11.9552 Tf 5.787 1.793 Td [( z = 1 2 @g z; 1 @ n ds z 2 \051 : .1.2 Here ds denotesarclengthmeasureand n istheoutwardnormalalong.Recall that g z; 1 istheuniqueharmonicfunctiononwhichvanishesontheboundary )-368(andsatises g z; 1 log z as z !1 .Ifisaconformalmappingofonto theexterioroftheunitdisk,thenbytheuniquenessofGreen'sfunction, g z; 1 = log j z j .Thus,byequation.1.2, d )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(= )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j 0 j ds .Forexample,if)-278(=[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] then d )]TJ/F19 11.9552 Tf 5.787 1.793 Td [( x = p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 dx .See[16,ch.3]forastellarintroduction. Equations.0.1cannowbeconsolidated: lim n !1 n n ;z = d d )]TJ/F19 11.9552 Tf 6.983 9.994 Td [( z ; .1.3 where)-345(istheunitcircleorinterval.SincetheChristoelfunctionsareinma,itis easytoshowthattheupperbound, limsup n !1 n n ;z d d )]TJ/F19 11.9552 Tf 6.982 9.994 Td [( z ; .1.4 holdsforallniteBorelmeasuresa.e.onthesupport.See[11,p.435].Until fairlyrecently,however,.1.3wasknownonlyforcontinuousweightsboundedaway fromzero.Thatitholdsalmosteverywherewithrespectto )]TJ/F19 11.9552 Tf 5.787 1.794 Td [(forthemoregeneral classofmeasuressatisfyingSzego'scondition, Z log d d )]TJ/F32 11.9552 Tf 6.982 26.851 Td [( d )]TJ/F27 11.9552 Tf 9.107 1.793 Td [(> ; .1.5 wasalong-standingconjecture,nallyprovedbyMate,Nevai,andTotikin1991 [11]. Inthischapter,weaimtoprove.1.3formeasuressupportedonsmoothcurves, )]TJ/F20 11.9552 Tf 10.8 0 Td [(2 C 1 ; ,whoseweightssatisfySzeg}o'scondition,.1.5.Weaccomplishthisbyan 36

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abstractionofMate'stechniquein[11]andinvestigatesomeapplications. 3.2MainResults Inwhatfollows, C denotethecomplexplane, U ,abounded,simplyconnecteddomain withboundary)-278(:= @U intheclass C 1 ; ;and:= C n U .denotestheclosedunit diskand R := f z : j z j R g withboundaries and R ,respectively.istheouter mappingfunctionof,aconformalmappingofonto C n with 1 = 1 and := )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 maps U conformallyand := )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .)]TJ/F28 7.9701 Tf 17.428 -1.793 Td [(R := R if R 1 and)]TJ/F28 7.9701 Tf 29.554 -1.793 Td [(R := R if R< 1. U R istheinteriorof)]TJ/F28 7.9701 Tf 91.785 -1.793 Td [(R and R := C n U R .Let ds denote arclengthmeasurealong)-285(andlet d = W j 0 j ds + d s beameasuresupportedon)]TJ -285.952 -20.922 Td [(withsingularpart s .Notethat d=d )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(=2 W almosteverywhere,sothefollowing, ourmainresult,isthecorrectabstractionofequation.1.3: Theorem3.2.1 Let d = W j 0 j ds + d s beapositive,Borelmeasuresupportedona closedcurve )]TJ/F20 11.9552 Tf 10.635 0 Td [(2 C 1 ; forsome > 0 andassumethat W satisesSzeg}o'scondition, .1.5.If W isboundedand d s 0 ,or > 1 = 2 ,then lim n !1 n n ; 0 = W 0 ; .2.6 for )]TJ/F46 11.9552 Tf 5.787 1.793 Td [(-almostevery 0 2 )]TJ/F46 11.9552 Tf 7.314 0 Td [(. Theorem3.2.1isreallytheintersectionoftwobroaderresults.Unfortunately,our proofoftheupperbound,.1.4,requiresamorestringentrestrictiononthesmoothnessof)-374(thanthatofthelowerbound.Thisinadequacyappearstobeintrinsicto ourmethod,aswillbecometransparentintheproof. Theorem3.2.2 Let d = W j 0 j ds + d s beapositive,Borelmeasuresupportedon aclosedcurve )]TJ/F20 11.9552 Tf 10.635 0 Td [(2 C 1 ; .If W isboundedand d s 0 ,or > 1 = 2 ,then limsup n !1 n n ; 0 W 0 ; .2.7 for )]TJ/F46 11.9552 Tf 5.787 1.794 Td [(-almostevery 0 2 )]TJ/F46 11.9552 Tf 7.314 0 Td [(. 37

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Theorem3.2.3 Let d = W j 0 j ds + d s beapositive,Borelmeasuresupportedon aclosedcurve )]TJ/F20 11.9552 Tf 10.635 0 Td [(2 C 1 ; where > 0 .If W satises.1.5then liminf n !1 n n ; 0 W 0 ; .2.8 for )]TJ/F46 11.9552 Tf 5.787 1.793 Td [(-almostevery 0 2 )]TJ/F46 11.9552 Tf 7.315 0 Td [(. Theorem3.2.2admitsthesimplerproof,sinceitisanupperboundontheinmum, n .Wesimplyneedasequenceofpolynomials, Q n z ,whose L 2 -normsconverge attheoptimalratedictatedbyTheorem3.2.3.Thisiseasyif)-295(istheunitcircle:We proveditalreadyinTheorem2.1.4usingthesequence Q n z = 1 n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 0 z k .2.9 Forgeneral,theupperboundcanbededucedfromthiscasebysubstituting z k in Q n z bythe k -thorder1-Faberpolynomialassociatedwith, F k z = 1 2 i Z )]TJ/F29 5.9776 Tf 5.288 -1.34 Td [(R k )]TJ/F27 11.9552 Tf 11.955 0 Td [(z 0 d; z 2 U R : .2.10 Let Q n z := 1 n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 F k z and S n z := 1 n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 0 z k z : .2.11 With d ~ := d e i ,wehave Z j S n j 2 d = Z 1 n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 k 2 j 0 j 2 d = Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( 1 n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 e ik 2 j 0 e i j )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 d ~ = 2 n n d ~ = j 0 j 2 ; 1 ; .2.12 andsotherevisedsequence, f Q n g ,willserveourpurposeifwecanshowthat Q n )]TJ/F27 11.9552 Tf -424.076 -20.922 Td [(S n 0withsucientrapidity.Wecan,atleastif > 1 = 2or W isbounded. 38

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Lemma3.2.4 Forany > 0 and )]TJ/F20 11.9552 Tf 10.635 0 Td [(2 C 1 ; Q n )]TJ/F27 11.9552 Tf 11.955 0 Td [(S n 0 pointwiseand Z )]TJ/F20 11.9552 Tf 7.78 10.793 Td [(j Q n )]TJ/F27 11.9552 Tf 11.955 0 Td [(S n j 2 ds = o n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 : .2.13 If > 1 = 2 thenthelimitholdsuniformlyfor z 2 )]TJ/F46 11.9552 Tf 7.314 0 Td [(: sup z 2 )]TJ/F20 11.9552 Tf 8.489 9.764 Td [(j Q n z )]TJ/F27 11.9552 Tf 11.955 0 Td [(S n z j = o n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = 2 : .2.14 Ourchoiceof Q n in.2.11isanaturalgeneralizationof.2.9:TheFaberpolynomialsgeneralizethemonomials, z k ,inavarietyofcontexts,e.g.locallyuniform taylor-seriestypeexpansionsofanalyticfunctionsin U .LetusquicklyverifyTheorem3.2.2usingtheseestimates. Proof. Theorem3.2.2Minkowskiigives p n k Q n k L 2 p n k Q n )]TJ/F27 11.9552 Tf 11.955 0 Td [(S n k L 2 + p n k S n k L 2 : .2.15 UndertheconditionsofTheorem3.2.2,Lemma3.2.4impliesthat k Q n )]TJ/F27 11.9552 Tf 12.167 0 Td [(S n k L 2 = o = p n sothersttermontheright-handsidevanishesas n !1 .Ourchoiceof theparticularconformalmappingin.2.11isarbitrary,sowemaypresumethat theFejermeansin.2.12convergeat z =1.Thus,.2.15impliesthat limsup n !1 n 2 Z j Q n j 2 d lim n !1 n 2 Z j S n j 2 d = W 0 j 0 0 j 2 ; where 0 =.Finally,since Q n 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(S n 0 0and S n 0 = 0 0 limsup n !1 1 2 n j Q n 0 j 2 Z j Q n j 2 d W 0 : .2.16 Themoredicultproofofthelowerbound,.2.8,willemployHardyspace methods. D z and Q z represent,respectively,theouterfunctionsassociatedwith log p W in U and.Theyareanalyticandnon-vanishingintheirrespectivedomains 39

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andhavenon-tangentiallimitsalmosteverywhereattheircommonboundary,,with j D j 2 = j Q j 2 = W ,foralmostevery 2 .Fix 0 2 ,aLebesguepointof D Q andlog W ;andchooseapolynomial, P n ,ofdegreeatmost n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1forwhich n ; 0 = 1 j P n 0 j 2 1 2 Z j P n j 2 d: P n isdenedonlyuptoamultiplicativeconstant,sowemayassumethat j P n 0 j =1 and,forthesamereason,that 0 =1.Ifnecessary,multiply Q byaconstantof unitmodulussothat Q 0 = D 0 Theproofofthelowerboundgiveninthelastchapterformeasuressupported onthecirclereliesheavilyonaknowledgeofthelocationsofthezerosof P n .Szeg}o provedthatthezerosof P n lieontheunitcircle;forgeneralsupports,noanalogous resultsexist.WecansurmountthisobstacletoouradaptationofMate'smethod bymeansofaweightedBernstein-Walshinequalitywhichpermitsanestimateofthe sequence fj P n jg invanishingneighborhoodsofthepointofevaluation, 0 .Thisisthe approachconceivedbyV.Totikandthecontentofournextresult. Lemma3.2.5 Fix c> 0 andassumethat W 1 satisesSzeg}o'scondition. j P n z j isboundeduniformlyforall j z )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 j
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sincetheirweightsmaybehighlyerratic,notamenabletothemethodsofclassical approximationtheory.Althoughourresultsdonotfullyextendhis,theydoimply thefollowingsup-normestimateforSzeg}oweightswhichfollowsimmediatelyfrom theTheoremand.0.3: Corollary3.3.1 Let satisfyallthehypothesesofTheorem3.2.1.Foralmostevery 0 2 )]TJ/F46 11.9552 Tf 7.314 0 Td [(, j p n ; 0 j = o p n : OperatorTheoreticFormulationandIll-PosedProblems Let H beaHilbertspacewithinnerproduct ; .Let y 2 H andlet A bealinear operatoron H .Theproblemofsolvingtheequation Ax = y iscalledill-posedif A isnotinvertible.Ill-posedproblemsobviouslyhavenosolutioningeneral,although stableapproximatesolutionsminimizing k Ax )]TJ/F27 11.9552 Tf 11.387 0 Td [(y k maybefoundbycertainrecursive algorithms.Wewillrestrictourdiscussiontothecaseinwhich A = 0 )]TJ/F27 11.9552 Tf 11.864 0 Td [(T ,where T isanormaloperatorwithspectrum T =)]TJ/F20 11.9552 Tf 30.933 0 Td [(2 C 1 ; and 0 2 .Atypicalapproach usesapproximantsoftheform x n = Q n T y where Q n isapolynomialofdegreeat most n .See,forexample,[1].Ideally, k 0 )]TJ/F27 11.9552 Tf 12.234 0 Td [(T x n )]TJ/F27 11.9552 Tf 12.234 0 Td [(y k! 0as n !1 .Whatis theoptimalrateofthisconvergence? TheanswerfollowsimmediatelyfromTheorem3.2.1:If f E g 2 )]TJ/F19 11.9552 Tf 9.117 1.793 Td [(denotesthespectralfamilyofprojectionsassociatedwith T ,then k 0 )]TJ/F27 11.9552 Tf 11.956 0 Td [(T x n )]TJ/F27 11.9552 Tf 11.955 0 Td [(y k 2 = Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [( Q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 2 d y ; where d y = W y j d j + d s := y;dE y .If0 = 2 )-254(then W y satisesSzego's conditionpreciselywhen y liesoutsidetheclosedspanof S y := f T k y g k 1 .Indeed, Szego'sclassicalresultformeasures d = W d ontheunitcircleis inf p 2 A ;p =0 Z j 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(p j 2 d =exp 1 2 Z log W d ; where A isthesetofanalyticfunctionsin.Aconformalmappinggeneralizes 41

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thistoanysimplyconnecteddomainwithboundary)]TJ/F20 11.9552 Tf 297.557 0 Td [(2 C 1 ; aslongas0 = 2 Applyingthespectraltheoremasabove,itfollowsthat y isisolatedfromthespanof S y preciselywhenlog W y isintegrable.Wenowhaveanequivalentformulationofour mainresult. Corollary3.3.2 LetTbeaboundednormaloperatoronaHilbertspace, H ,with spectrum )]TJ/F20 11.9552 Tf 10.635 0 Td [(2 C 1 ; and 0 = 2 )]TJ/F46 11.9552 Tf 7.315 0 Td [(.If y 2 H n S y then liminf n !1 n k 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(T x n )]TJ/F27 11.9552 Tf 11.955 0 Td [(y k 2 W y 0 ; foralmostevery 0 2 )]TJ/F46 11.9552 Tf 7.314 0 Td [(.Equalityholdsif Q n isgivenby.2.11and > 1 = 2 ,or W y isboundedand d s 0 3.4Proofs WebeginwiththeproofofLemma3.2.5.Let D denoteharmonicmeasureinthe domain D .If @D issmooth,wehavetherepresentation d! D z; = K D z; j d j ,for 2 @D .Let r r< 1denoteaconformalmapof r onto C n with r 0 > 0and r z z z !1 andlet r denoteitsinverse.Since)]TJ/F20 11.9552 Tf 142.832 0 Td [(2 C 1+ f 0 r g Lip withuniformlyboundedlipschitzconstantsfor1 = 2 0 suchthatforall 1 = 2
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forall 2 )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(r and 2 r .Similarly,forall 2 )-326(and z 2 r K U ; C 2 1 )-222(j j 2 j )]TJ/F19 11.9552 Tf 11.955 0 Td [( j 2 : Notethat1 )-235(j j 2 dist )]TJ/F27 11.9552 Tf 11.866 0 Td [(; )]TJ/F28 7.9701 Tf 7.314 -1.794 Td [(r j r j 2 )]TJ/F19 11.9552 Tf 12.108 0 Td [(1,uniformlyfor 2 )]TJ/F28 7.9701 Tf 7.314 -1.794 Td [(r and 2 andforall1 = 2 1, 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( R 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j Re i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e it j 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e it j 2 dt = 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( 1 X n = R j n j e in t )]TJ/F28 7.9701 Tf 6.586 0 Td [( 1 X k = j k j e ik t )]TJ/F28 7.9701 Tf 6.586 0 Td [( dt = 1 X n = R j n j e in )]TJ/F28 7.9701 Tf 6.586 0 Td [( = 2 R 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j R )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( j 2 ; whichequalsthefractionontherighthandsideof.4.17. Proof. Lemma3.2.5Let h = h r representthesolutiontothedirichletproblemin r withboundarydata h =log j D j 2 )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(r .Firstweprovethatif r = r n < 1 43

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and1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r n 1 =n then j P n z j Ce )]TJ/F28 7.9701 Tf 6.586 0 Td [(h z + ng r z; 1 ; z 2 r ; .4.18 forsome C> 0independentof r and n .For z 2 )]TJ/F28 7.9701 Tf 7.314 -1.794 Td [(r and r<< 1, j P n z D z j 2 = 1 2 Z )]TJ/F29 5.9776 Tf 5.289 -0.997 Td [( P n 2 D 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(z d 1 2 Z )]TJ/F29 5.9776 Tf 5.288 -0.997 Td [( j P n D j 2 ds max 2 )]TJ/F29 5.9776 Tf 5.288 -0.996 Td [( 1 j )]TJ/F27 11.9552 Tf 11.955 0 Td [(z j : Letting 1 )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [(,wendthat j P n z D z j 2 max 2 )]TJ/F20 11.9552 Tf 10.573 7.527 Td [(j 0 j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 1 2 Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j P n j 2 W j 0 j ds C 1 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r C 2 j P n 0 j 2 n ; 0 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(r C 3 j P n 0 D 0 j 2 n )]TJ/F27 11.9552 Tf 11.955 0 Td [(r ; by.2.16.Theconstant C 3 isindependentof n and r .So, j P n z D z j isbounded uniformlyfor z 2 )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(r as r = r n 1aslongas1 )]TJ/F27 11.9552 Tf 12.235 0 Td [(r 1 =n .Thisproves,underthe statedassumptions,that u z =log j P n z j + h z )]TJ/F27 11.9552 Tf 11.955 0 Td [(ng r z; 1 isboundedon)]TJ/F28 7.9701 Tf 82.618 -1.793 Td [(r .Itisalsoclearlysubharmonicin r andboundedat 1 ,soitmust betruethat u z log C in r ,whichestablishes.4.18. Let'sexaminetheexponentontherightsideof.4.18.Since 0 r areuniformly bounded,wemaychoose > 0sothat S r := f w : j w )]TJ/F27 11.9552 Tf 12.617 0 Td [( 0 j < dist 0 ; )]TJ/F28 7.9701 Tf 7.315 -1.793 Td [(r g has diam r S r < 1 2 dist 0 ; )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(r forall1 = 2
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boundedon S r .Toseethis,let z 2 S r andapply.4.17toobtain )]TJ/F27 11.9552 Tf 11.955 0 Td [(h z = Z )]TJ/F29 5.9776 Tf 5.289 -0.997 Td [(r log j D j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K r z; j d j = Z )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(r Z )]TJ/F19 11.9552 Tf 7.779 10.793 Td [(log j D j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K U ; j d j K r z; j d j = Z )]TJ/F19 11.9552 Tf 7.779 10.793 Td [(log j D j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Z )]TJ/F29 5.9776 Tf 5.288 -0.996 Td [(r K U ; K r z; j d j j d j C 1 Z )]TJ/F19 11.9552 Tf 7.78 10.793 Td [(log j D j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j r j 2 j r z j 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j r r z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 2 j d j .4.19 Nowlet R r e i = r andset r z = e i and F r =log j D r R r e i j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Withachangeofvariables,wendthat )]TJ/F27 11.9552 Tf 11.955 0 Td [(h z C 2 Z F r 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 = R r 2 j e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i = R r j 2 d C 3 1 2 Z F r 1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 = R 0 r 2 j e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i = R 0 r j 2 d; where R 0 r > 1areconstants.Thesecondinequalityfollowsfromthefactthat R r isboundedawayfrom1uniformlyfor z 2 S r .ButthelasttermisthePoisson integral,[ PF r ] e i = R 0 r of F r evaluatedatthepoint e i = R 0 r =1 =R 0 r r z .Since r S r ,thispointiscontainedinthereciprocalsector, )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 isbased at1andnon-tangentialto @ ,sowemayapplyafundamentalinequalityforPoisson integralsofnitemeasuressee[17,p.242]toconcludethat )]TJ/F27 11.9552 Tf 9.298 0 Td [(h z C 4 MF r z 2 S r : MF r istheHardymaximalfunctionof F r : MF r =sup I 1 j I j Z I F r d C 5 sup I 1 j I j Z r I 0 log j D j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j d j = C 6 sup I j r I 0 j j I j 1 j r I 0 j Z r I 0 log W )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 j d j 45

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Heretheintervals I arecenteredat =0, I 0 = f e i : 2 I g and jj denotesarc lengthmeasure.Thelastintegralisboundedindependentlyof r sincethearcs r I shrinkto 0 ,whichisalebesguepointoflog W .Thisprovesourclaim. Finally,since z 2 S r impliesthat j r z j < 1+ c j 1 )]TJ/F27 11.9552 Tf 11.928 0 Td [(r j g r z; 1 =log j r z j c j 1 )]TJ/F27 11.9552 Tf 12.168 0 Td [(r j and,therefore, ng r z; 1 isboundedon S r as n !1 since1 )]TJ/F27 11.9552 Tf 12.168 0 Td [(r n 1 =n So,.4.18ensuresthat P n z isboundeduniformlyonthesets S r .Forsuciently small > 0andlarge n r n =1 )]TJ/F19 11.9552 Tf 12.75 0 Td [(1 =n makes f z : j z )]TJ/F27 11.9552 Tf 12.75 0 Td [( 0 j 1 g insucha waythat j z n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 1 =n .Wewillprovethat limsup n !1 8 < : j P n z n Q z n j)-222(j P n 0 j p n 0 n X k =0 j z n j 2 k 1 = 2 9 = ; 0 : .4.20 Letderive.2.8from.4.20.Itsucestoshow,forarbitrarilysmall > 0 > 0 and > 0,theexistenceofasequence z n 1non-tangentiallyin f z : j z j > 1 g and with 0 =n< j z n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j <=n suchthat liminf n !1 j P n z n j )]TJ/F27 11.9552 Tf 11.955 0 Td [( j P n 0 j : .4.21 Forthen z n 0 nontangentiallyin,so j Q z n j 2 W 0 whichimplies byvirtueof.4.20that liminf n !1 n ; 0 n X k =0 j z n j 2 k )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 W 0 : This,togetherwith n X k =0 j z n j 2 k n 1+ n 2 n
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proves.2.8.Now,Lemma3.2.5providesconstants M and c suchthat j P n z j M whenever j z )]TJ/F27 11.9552 Tf 11.956 0 Td [( 0 j c=n .Thus, j z )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 j c 2 n = j P 0 n z j = 1 2 i Z j )]TJ/F28 7.9701 Tf 6.586 0 Td [(z j = c= 2 n P n z )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 d 2 M c n; soif issucientlysmalland j z )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 j <=n then j P n z )]TJ/F27 11.9552 Tf 11.955 0 Td [(P n 0 j = Z z 0 P 0 n d and,consequently, j P n z j )]TJ/F27 11.9552 Tf 12.861 0 Td [( j P n 0 j .Since j 0 j isuniformlyboundedthe requiredsequenceexists. Toestablish.4.20,denethekernels H n z := n X k =0 z k n z k : With e F := F ,wehave 1 2 i Z )]TJ/F27 11.9552 Tf 7.779 10.793 Td [(P n Q H n 0 d = 1 2 i Z e P n w e Q w 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n +1 n w n +1 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n w dw = 1 2 i Z e P n w e Q w 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n w dw )]TJ/F27 11.9552 Tf 13.15 8.088 Td [(z n +1 n 2 i Z e P n w e Q w w n w )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n dw: e P n w n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 as w !1 ,sothesecondintegralgives,withthesubstitution z =1 =w z n n 2 i Z e P n =z e Q =z z n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 z )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 n )]TJ/F27 11.9552 Tf 11.955 0 Td [(z dz = )]TJ/F27 11.9552 Tf 9.298 0 Td [(z n e P n z n e Q z n : Subtracting,weobtain )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n e P n z n e Q z n + 1 2 i Z )]TJ/F27 11.9552 Tf 7.78 10.793 Td [(P n Q H n 0 d = 1 2 i Z e P n w e Q w 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n w dw =: I: .4.22 47

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Weshowthat I 0as n !1 .Tothisend,choosealarge a> 0andlet K 1 = [ )]TJ/F27 11.9552 Tf 9.298 0 Td [(a=n;a=n ]and K 2 =[ )]TJ/F27 11.9552 Tf 9.299 0 Td [(; ] n K 1 .With I j := 1 2 i Z K j e P n w e Q w 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n w dw; w = e i I = I 1 + I 2 .Since z n 1non-tangentially, j e i )]TJ/F27 11.9552 Tf 11.168 0 Td [(z n j r ,forsome r> 0.Therefore, j I 2 j 2 1 2 Z K 2 j e P n e Q j 2 d 1 2 Z K 2 d j e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n j 2 1 2 Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j P n Q j 2 j d j 1 r 2 Z 1 a=n d 2 = 1 2 Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j P n j 2 W j 0 j ds n r 2 a = j P n 0 j 2 n 0 n ar 2 ; .4.23 whichcanbemadearbitrarilysmallforsucientlylarge a ,since n n 0 isbounded forall n ,by3.2.16.Toestimate I 1 ,decomposeitas I 1 = I 11 + I 12 ,where I 11 := 1 2 i Z K 1 e P n w e D w 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n w dw and I 12 := 1 2 i Z K 1 e P n w 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n w h e Q w )]TJ/F32 11.9552 Tf 14.307 3.022 Td [(e D w i dw: Lemma3.2.5impliesthat j I 12 j 1 2 max w 2 K 1 e P n w 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n w Z K 1 j e Q e i )]TJ/F32 11.9552 Tf 14.306 3.022 Td [(e D e i j d C j z n j)]TJ/F19 11.9552 Tf 17.933 0 Td [(1 Z K 1 j e Q e i )]TJ/F32 11.9552 Tf 14.307 3.022 Td [(e D e i j d; whichtendsto0as n !1 since n j z n j)]TJ/F19 11.9552 Tf 17.043 0 Td [(1 > 0 > 0;1isaLebesguepointof e Q and e D ;and e Q = e D .With J 1 := K 1 ,wehave I 11 = 1 2 i Z K 1 e P n w e D w w w )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n dw = 1 2 i Z J 1 P n D )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n 0 d: Consider G z := 0 + 0 0 z )]TJ/F27 11.9552 Tf 11.956 0 Td [( 0 ,thelinearizationofabout 0 ,anddene I 11 := 1 2 i Z J 1 P n D G )]TJ/F27 11.9552 Tf 11.956 0 Td [(z n 0 d: 48

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Since 2 C 1 ; j G )]TJ/F19 11.9552 Tf 11.961 0 Td [( j C 1 j )]TJ/F27 11.9552 Tf 11.96 0 Td [( 0 j 1+ .Consequently,thecurve L := f G : 2 )]TJ/F20 11.9552 Tf 7.315 0 Td [(g istangentto at1so,since z n 1non-tangentiallyto j G )]TJ/F27 11.9552 Tf 12.591 0 Td [(z n j c j )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n j c 0 =n forall 2 )-326(andsucientlylarge n .Thus, 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n )]TJ/F19 11.9552 Tf 35.244 8.088 Td [(1 G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n C 2 j )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 j 1+ n )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 C 3 n 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( ; for 2 J 1 .ByapplyingCauchy'sinequality,weobtainthefollowing: j I 11 )]TJ/F27 11.9552 Tf 11.955 0 Td [(I 11 j 1 2 Z J 1 P n D 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n )]TJ/F19 11.9552 Tf 35.244 8.088 Td [(1 G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n 0 d + 1 2 Z J 1 P n D G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n [ 0 )]TJ/F19 11.9552 Tf 11.956 0 Td [( 0 ] d 1 2 Z J 1 j P n D j 2 j 0 jj d j 1 = 2 Z J 1 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n )]TJ/F19 11.9552 Tf 35.245 8.088 Td [(1 G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n 2 j 0 d j 1 = 2 + C 4 n 1 2 Z J 1 j P n D j 2 j 0 jj d j 1 = 2 Z J 1 j 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [( 0 j 2 j 0 j j d j 1 = 2 .4.24 Thersttermontherightsideof.4.24isboundedaboveby C 5 p j P n 0 j n ; 0 n 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( j J 1 j 1 = 2 ; which,byLemma3.2.5,convergesto0as n !1 jj denotesarc-lengthmeasure. Thenaltermdoesaswell,sinceitslastintegrandiscontinuous.Now,theintegrand of I 11 isholomorphicin U ,sowemaydeformthecontour, J 1 ,tothehomologous contour, )]TJ/F27 11.9552 Tf 9.299 0 Td [(J 2 := )]TJ/F19 11.9552 Tf 9.299 0 Td [( K 2 .Thisgives j I 11 j = 1 2 i Z J 2 P n D G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n 0 d 1 2 Z )]TJ/F20 11.9552 Tf 7.78 10.793 Td [(j P n D j 2 j 0 jj d j 1 = 2 Z J 2 j 0 j j G )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n j 2 j d j 1 = 2 C 6 Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j P n D j 2 j 0 jj d j 1 = 2 Z K 2 d j e i )]TJ/F27 11.9552 Tf 11.955 0 Td [(z n j 2 1 = 2 : 49

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Asin.4.23,thiscanbemadearbitrarilysmallbychoosingsucientlylarge a .We concludethat I 1 and I 2 arearbitrarilysmallif n issucientlylarge,whichestablishes that I 0as n !1 .Finally, 1 2 i Z )]TJ/F27 11.9552 Tf 7.779 10.793 Td [(P n Q H n d 2 1 2 Z )]TJ/F20 11.9552 Tf 7.779 10.793 Td [(j P n Q j 2 j d j 1 2 Z )]TJ/F20 11.9552 Tf 7.78 10.793 Td [(j H n j 2 j d j = j P n 0 j 2 n 0 1 2 Z )]TJ/F28 7.9701 Tf 6.586 0 Td [( n X k =0 z k n e )]TJ/F28 7.9701 Tf 6.586 0 Td [(ik 2 d = j P n 0 j 2 n 0 n X k =0 j z n j 2 k : This,togetherwith.4.22,completestheproof. Proof. Lemma3.2.4 TheLaurentseries c )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + c 0 + c 1 = + convergesuniformlyon)]TJ/F28 7.9701 Tf 130.567 -1.793 Td [(R for sucientlylarge R ,so 1 2 i Z )]TJ/F29 5.9776 Tf 5.288 -1.34 Td [(R 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z d = c )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z 2 \051 ; and F n z = I + c )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 n z ,where I := 1 2 i Z )]TJ/F29 5.9776 Tf 5.289 -1.34 Td [(R n )]TJ/F19 11.9552 Tf 11.955 0 Td [( n z )]TJ/F27 11.9552 Tf 11.956 0 Td [(z 0 d: Makethesubstitutions w = e it = and e i = z andfactortheintegrandto obtain I = 1 2 i Z R w )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i w )]TJ/F19 11.9552 Tf 11.955 0 Td [( e i n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 w k e i n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(k dw = 1 2 Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( e it )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i e it )]TJ/F19 11.9552 Tf 11.955 0 Td [( e i n X k =1 e ikt e i n )]TJ/F28 7.9701 Tf 6.586 0 Td [(k dt .4.25 50

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Sincethelaterintegrandiscontinuousandanalyticin j w j > 1,wemaydeformthe contourbackto .Nowlet f x;y := e ix )]TJ/F27 11.9552 Tf 11.955 0 Td [(e iy e ix )]TJ/F19 11.9552 Tf 11.955 0 Td [( e iy and F y x := f x;y ; andlet S n F denotethepartialsumsoftheFourierseriesof F and D n t theDirichlet kernels. F isthecontinuationto ofafunctionanalyticandboundedin C n ,soits Fouriercoecients, ^ F k ,withpositiveindexvanish.Thus,from.4.25weobtain I = e in 1 2 Z F t n X k =1 e ik t )]TJ/F28 7.9701 Tf 6.587 0 Td [( dt = e in Z F t D n t )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 dt = e in )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(S n F )]TJ/F19 11.9552 Tf 15.16 3.022 Td [(^ F and,consequently, F n z = e in S n F )]TJ/F19 11.9552 Tf 15.635 3.022 Td [(^ F + c )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .Inthefollowing,themeasures dw=w j R areuniformlyboundedandtheintegrandisanalyticfor j w j > 1and continuousfor j w j 1sowemaydeformthecontourofintegrationandevaluate asymptotically:since w c )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 w as w !1 ^ F = 1 2 i Z w )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i w )]TJ/F19 11.9552 Tf 11.955 0 Td [( e i dw w =lim R !1 1 2 i Z R w w dw w = c )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : Furthermore,since F = 0 e i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = 0 z ,wehave F n z )]TJ/F19 11.9552 Tf 11.956 0 Td [( 0 z n z = e in S n F )]TJ/F27 11.9552 Tf 11.956 0 Td [(F andso Z n X k =0 F k z )]TJ/F19 11.9552 Tf 11.956 0 Td [( 0 z k z 2 ds Z )]TJ/F28 7.9701 Tf 6.587 0 Td [( n X k =0 e ik )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(S k F )]TJ/F27 11.9552 Tf 11.955 0 Td [(F 2 d: .4.26 Thisreducestheproblemtoanestimationtheconvergence S n F F Weshowthatthefunctions F 2 Lip andhaveuniformlyboundedLipschitz constants.Since F areuniformlybounded,itsucestoprovethistrueofthefamily 51

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f 1 =F g .Set e i k = u k and e i = w .If u k 6 = w ,wehave F 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(F 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = 1 u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w Z u 1 w 0 z dz )]TJ/F19 11.9552 Tf 27.572 8.087 Td [(1 u 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w Z u 2 w 0 z dz = 1 u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w Z u 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(w 0 )]TJ/F19 11.9552 Tf 5.48 -9.683 Td [( 0 z + w )]TJ/F19 11.9552 Tf 11.955 0 Td [( 0 z=a + w dz; where a = u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w = u 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w .Thus, j F 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(F 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j C j u 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(w j Z u 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(w 0 j z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z=a j j dz j C 0 j 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 =a j j u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w j j u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(w j 1+ = C 0 j u 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(u 2 j : If u 1 = w ,theresultfollowsinasimilarway.Astandardresultfromapproximation theorynowapplies: j S n F x )]TJ/F27 11.9552 Tf 11.667 0 Td [(F x j Cn )]TJ/F28 7.9701 Tf 6.586 0 Td [( ln n ,where C isindependentof x and .Seee.g.[8,pp.180,192-194].Thus, 1 n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X k =0 j S k F )]TJ/F27 11.9552 Tf 11.955 0 Td [(F j C n n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X k =0 ln k k = O ln n n ; whichproves.2.14if > 1 = 2. Toprove.2.13,considerthepartialsums, S nm ,ofthedoubleFourierseriesof f x;y .Weclaimthattheyconvergeto f uniformlyin x and y as n m !1 .Let D n t denotethenormalizedDirichletkernels. S nm x;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x;y = ZZ D n t D m u [ f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(u )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x;y ] dtdu = ZZ D n t D m u [ f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(u )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;y ] dtdu + Z D n t [ f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x;y ] dt =: I 1 + I 2 : 52

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Wehavealreadyshownthat j I 2 j = O n )]TJ/F28 7.9701 Tf 6.586 0 Td [( ln n .Ontheotherhand,since F x y = F y x j I 1 j sup t Z D m u f x )]TJ/F27 11.9552 Tf 11.956 0 Td [(t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(u )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;y du Z j D n j d =sup t j S m F x )]TJ/F28 7.9701 Tf 6.587 0 Td [(t y )]TJ/F27 11.9552 Tf 11.955 0 Td [(F x )]TJ/F28 7.9701 Tf 6.586 0 Td [(t y j ln n C ln n ln m m : Thisprovesourclaim. Wemaynowevaluatetherighthandsideof.4.26intermsofthedevelopment w )]TJ/F27 11.9552 Tf 11.955 0 Td [(u w )]TJ/F19 11.9552 Tf 11.955 0 Td [( u X m;l c ml w m u l :=lim N !1 N X m;l =0 c ml w m u l j w j 1 ; j u j 1 ; whichconvergesuniformlyonthetorus S k F = X j j j k X m 0 c mj e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i m + j )]TJ/F28 7.9701 Tf 6.586 0 Td [(k 2 d = n X k =0 1 X r = j )]TJ/F28 7.9701 Tf 6.587 0 Td [(k X j>k j c r + k )]TJ/F28 7.9701 Tf 6.586 0 Td [(j;j j 2 = n X k =0 1 X r = l 1 X l =1 j c r )]TJ/F28 7.9701 Tf 6.586 0 Td [(l;l + k j 2 = n X k =0 1 X r =0 1 X l =1 j c r;l + k j 2 : Theseseriesconvergesincethecontinuityof f impliesthat X r;l j c r;l j 2 = 1 4 2 ZZ j f x;y j 2 dxdy< 1 : Therefore, 1 n n X k =0 X r;l j c r;l + k j 2 0 n !1 ; which,byvirtueof.4.26,thiscompletestheproof. 53

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References [1]A.BakushinskyandA.Goncharsky. Ill-PosedProblems:TheoryandApplications .KluwerAcademicPublishers,Boston,1994. [2]E.Findley. FineAsymptoticsofChristoelFunctionsforGeneralMeasures preprint. [3]E.Findley. UniversalityforLocallySzeg}oMeasures. J.Approx.Theory.155 ,pp.136-154. [4]G.Freud. OrthogonalPolynomials .PergamonPress,Oxford,1971. [5]U.GrenanderandG.Szeg}o. ToeplitzFormsandtheirApplications .Unviersityof CaliforniaPress,LosAngeles,1958. [6]L.Golinskii.TheChristoelFunctionforOrthogonalPolynomialsonaCircular Arc. J.Approx.Theory 101 ,pp.165-174 [7]K.Homan, BanachSpacesofAnalyticFunctions ,PrenticeHall,London1962. [8]N.Korneichuk. ExactConstantsinApproximationTheory. CambridgeUniversity Press,NewYork,1991. [9]E.Levin,D.Lubinsky.ApplicationsofUniversalityLimitstoZerosandReproducingKernelsofOrthogonalPolynomials,toappearin J.Approx.Theory http://www.math.gatech.edu/lubinsky/SelectedPapers.html 54

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[10]D.S.Lubinsky.ANewApproachtoUniversalityLimitsInvolvingOrthogonal Polynomials,toappearin Ann.ofMath. http://www.math.gatech.edu/lubinsky/SelectedPapers.html [11]A.Mate,P.NevaiandV.Totik.Szeg}o'sExtremumProblemontheUnitCircle, Ann.ofMath. 134 ,433{453. [12]A.MateandP.Nevai.Bernstein'sinequalityin L p for0
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[23]J.L.Ullman.OntheRegularBehaviorofOrthogonalPolynomials. Proc.London Math.Soc. 24 :119-48. [24]A.Zygmund, TrigonometricSeries ,Vols.IandII,2nded.,CambridgeUniv. Press,Cambridge,1979. 56

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AbouttheAuthor ElliotM.FindleycompletedaB.S.inPhysicsattheUniversityofSouthFloridain 2003andanM.A.inAppliedMathematicsatUSFin2004.RaisedinBakerseld, California,hegraduatedfromBakerseldChristianHighSchoolin1998.Laterthat year,hemovedtoTempleTerrace,Florida,toattendFloridaCollegefortwoyears beforetransferringtoUSF. Hisscholarlyinterestsincludeallareasofanalysis,butespeciallyapproximation theoryandgeneralorthogonalpolynomials.