Bernstein型不等式与Steckin型不等式

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1、万方数据第8卷第3期2009年5月杭州师范大学学报(自然科学版)JournalofHangzhouNormalUniversity《NaturalScienceEdition)V01．8No．3May2009ArticleID：1674—232X(2009)03—0173—07Bernstein—TypeandSteckin—TypeInequality(I)LINShao—b01，CAOFei—lon92+(1．CollegeofScience，HangzhouNormalUniversity，Hangzhou310036，China；2．Departme

2、ntofInJormationandMathematicsSciences-ChinajiliangUniversity，Hangzhou，310018，China)Abstract：BernsteinandSteckininequalitiesplayanimportantroleinapproximationtheory．Forexample，BernsteininequalityisthemaintooltoprovetheInverseTheoremandEmbeddingTheorem．Thispaperestablishestherelation

3、betweenweightedBernstein—typeinequalityandSteckin-typeinequalityinageneralsystemofmultivariatefunction．Keywords：Bernstein-typeinequality；Steckin-typeinequality)moduliofsmoothnessCLCnumher：0174．41MSC2000：41A17Articlecharacter：A1IntroductionInthispaper，wealwaysdenoteby竹anon—negativei

4、nteger，andbyCanabsolutepositiveconstant．WealsouseC(a1，a2，a3，⋯)todenoteapositiveconstantdependingonlyona。(i=1，2，⋯)．Let纪and咒denotethesetofallalgebraicandtrigonometricpolynomialsofdegreeatmost卵withrealcoefficientsrespectively．WedenotebySameasurablesetinR4．ByL9(S)，wedenotethesetofallco

5、ntinuousrealvaluedfunctionsandP—integrablefunctionsonSendowedwiththenormsJ厂lI。。．-

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9、L一(s)：=esssupl厂(z)Iz∈5and，：=II川以。，一{．『。i厂(z"妇r<∞，1≤夕<。。．ForthesakeofbrevitywesetDk(f)一希％，where志l+⋯+志d=忌．ThenwedenotebyW；(S)，r=1，2，⋯，thecollectionoffunctionsforwhichD^一1(／)isabsolutelycontinuousand

10、D^(厂)∈L’(S)．Receiveddate：2009—01—18Foundationitem：SupportedbytheNationalNaturalScienceFoundationofChina(No．60873206)，andtheNaturalScienceFoundationofZhejiangProvinceofChina(No．Y．7080235)．Biography：LinShaobo(1983一)，male。borninWenzhou，ZhejiangProvince，graduate，majoredinapproximationt

11、heory．’Correspondingauthor：CaoFeilong(1965一)。male，borninTaizhou，ZhejiangProvince，professor，majoredinapproximationtheoryandneuralnetworks．E—mail：feilongcao@gmail．tom万方数据174杭州师范大学学报(自然科学版)2009年Asearlyas1911，Bernsteininhisdoctoraldissertation[1]provedthefollowingwell—knowninequality1l

12、T’。11c[一，，力≤nllT。llc[一。司，T