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1、NApproximationinCNormLevenberg3November2006Abstract.Thisisasurveyarticleonselectedtopicsinapproximationtheory.Thetopicseitherusetechniquesfromthetheoryofseveralcomplexvariablesorariseinthestudyofthesubject.Thesurveyisaimedatreadershavinganacquaintancewithst
2、andardresultsinclassicalapproximationtheoryandcomplexanalysisbutnoaprioriknowledgeofseveralcomplexvariablesisassumed.MSC:32-02,41-021Introductionandmotivation.........922Polynomialhullsandpolynomialconvexity....963PlurisubharmonicfunctionsandtheOka-Weiltheo
3、rem..974QuantitativeapproximationtheoremsinC....1035TheBernstein-WalshtheoreminCN,N>1....1056QuantitativeRunge-typeresultsinmultivariateapproximation1097Mergelyanpropertyandsolving∂¯.......1118Approximationontotallyrealsets......1159Lagrangeinterpolationand
4、orthogonalpolynomials...11810Kergininterpolation..........12111RationalapproximationinCN........12512Markovinequalities..........12813Appendixonpluripolarsetsandextremalpshfunctions.13014AppendixoncomplexMonge-Amp`ereoperator...13415Afewopenproblems........
5、..135References.............1361IntroductionandmotivationarXiv:math/0611249v1[math.CA]8Nov2006NNLetC={(z1,...,zN):zj∈C}wherezj=xj+iyjandidentifyR={(x1,...,xN):xj∈R}.NAcomplex-valuedfunctionfdefinedonanopensubsetofCisholomorphicifitisseparatelyholomorphicinth
6、eappropriateplanarregionasafunctionofonecomplexvariablewheneachoftheremainingN−1variablesarefixed.Thisdeceptivelysimple-mindedcriterionisequivalenttoanyotherstandarddefinition;e.g.,fislocallyrepresentablebyaconvergentpowerseriesinthecomplexcoordinates;orfisof
7、classC1andsatisfiestheCauchy-Riemannsystem∂f1 ∂f∂f:=+i=0,j=1,...,N.∂z¯j2∂xj∂yjSurveysinApproximationTheory92Volume2,2006.pp.92–140.Copyrightoc2006SurveysinApproximationTheory.ISSN1555-578XAllrightsofreproductioninanyformreserved.NApproximationinC93Inparticu
8、lar,holomorphicfunctionsaresmooth,indeed,real-analytic;whereastheseparatelyholomorphiccriterionmakesnoaprioriassumptiononcontinuity(Hartogsseparateanalyticitytheorem,circa1906;cf.,[Sh]section6).Wemaken
