INTEGRAL MODIFICATION OF CERTAIN POSITIVE LINEAR OPERATORS

Abstract

This thesispresentsapproximationoffunctionsbyseveralwellknownpositivelin- ear operators,bytheirgeneralizedformsandintegralmodi cations.Wedividethe thesis intoninechapters.Chapter0isanintroductorypartofthethesiswhichdeals with theupbringingofapproximationtheory,literaturesurvey,somenotationsand basic de nitionsofapproximationmethodswhichareusedthroughoutthethesis. In the rstchapter,wede neagenuinefamilyofBernstein-Durrmeyertype operatorsbasedonPolyabasisfunctions.Weestablishaglobalapproximationthe- orem, localapproximationtheorem,Voronovskaya-typeasymptotictheoremanda quantitativeestimateofthesametype.Lastly,westudytheapproximationoffunc- tions havingaderivativeofboundedvariation. The secondchapterisacontinuationofthe rstoneinwhichweintroducethe B eziervariantofgenuineDurrmeyertypeoperatorsandgivedirectapproximation results andaVoronovskayatypetheorembyusingtheDitzian-Totikmodulusof smoothness.Therateofconvergenceforfunctionswhosederivativesareofbounded variationisalsoobtained.Further,weshowtherateofconvergenceoftheseopera- tors tocertainfunctionsbyillustrativegraphicsusingtheMatlabalgorithms. In thethirdchapter,wede netheSz asz-Durrmeyertypeoperatorsbymeansof i multipleAppellpolynomials.WestudyaquantitativeVoronovskayatypetheorem and Gr uss-Voronovskayatypetheorem.Wealsoestablishalocalapproximation theorem intermsoftheSteklovmeansandVoronovskayatypeasymtotictheorem. Further,wediscussthedegreeofapproximationbymeansofaweightedspace. Lastly,we ndtherateofapproximationoffunctionshavingderivativesofbounded variation. In thefourthchapter,weintroducetheB eziervariantofDurrmeyermodi cation of theBernsteinoperatorsbasedonafunction : Wegivetherateofapproximation of theseoperatorsintermsofusualmodulusofcontinuityandthe K􀀀functional. Next, weestablishthequantitativeVoronovskajatypetheorem.Inthelastsection weobtaintherateofconvergenceforfunctionshavingderivativesofboundedvari- ation. In the fthchapter,wede neasequenceofStancutypeoperatorsbasedonthe same function as de nedintheprecedingchapterandshowthattheseoperators presentabetterdegreeofapproximationthantheoriginalones.Wegiveadirect approximationtheorembymeansoftheDitzian-Totikmodulusofsmoothnessand a Voronovskayatypetheorem. In thesixthchapter,weintroducetheB eziervariantofmodi edSrivastava- Gupta operatorsandgiveadirectapproximationtheorembymeansoftheDitzian- Totikmodulusofsmoothnessandtherateofconvergenceforfunctionswithderiva- tivesequivalenttoafunctionofboundedvariation.Furthermore,weshowthe comparisons oftherateofconvergenceoftheSrivastava-Guptaoperatorsvis-a-vis its B eziervarianttoacertainfunctionbyillustrativegraphicsusingMaplealgo- rithms. ii In theseventhchapter,weconstructtheStancu-Durrmeyer-typemodi cationof q-Bernstein operatorsbymeansofJacksonintegral.Here,weestablishbasiccon- vergencetheorem,localapproximationtheoremandanapproximationresultfora Lipschitztypespace.Also,weestablishtheKorovkintype A-statistical approxi- mation theoremandratesof A-statistical convergenceintermsofthemodulusof continuity. The lastchapterisancontinuationofourworkinchapterseven.Here,wecon- struct abivariategeneralizationofStancu-Durrmeyertypeoperatorsandstudythe rate ofconvergencebymeansofthecompletemodulusofcontinuityandthepartial moduliofcontinuity.Subsequently,wede netheGBS(GeneralizedBooleanSum) operatorsofStancu-Durrmeyertypeandgivetherateofapproximationbymeans of themixedmodulusofsmoothnessandtheLipschitzclassofB ogel-continuous functions.