STUDY ON COVERGENCE OF CERTAIN LINEAR POSITIVE OPERATORS

Abstract

In thethesis,westudyapproximationpropertiesofsomewellknownoperatorsandtheir q-analogues. Wedividethethesisintoeightchapters.Thechapter0includestheliterature survey,basicdefinitionsandsomebasicnotationsofapproximationmethodswhichwill be usedthroughoutthethesis. In thefirstchapter,wediscussedtheSchurertype q-Bernstein Kantorovichoperatorwhich wasintroducedbyLin.Weobtainalocalapproximationtheoremandthestatisticalcon- vergenceoftheseoperators.Wealsostudytherateofconvergencebymeansofthefirst order modulusofcontinuity,Lipschitzclassfunction,themodulusofcontinuityofthe first orderderivativeandtheVoronovskajatypetheorem. The secondchapterisconcernedwiththeStancu-Kantorovichoperatorsbasedon P´olya-Eggenbergerdistribution.Weobtainsomedirectresultsfortheseoperatorsby means oftheLipschitzclassfunction,themodulusofcontinuityandtheweightedspace. Also, westudyanapproximationtheoremwiththeaidoftheunifiedDitzian-Totikmodu- lus ofsmoothness ! (f; t); 0 1 and therateofconvergenceoftheoperatorsfor the functionshavingaderivativewhichislocallyofboundedvariationon [0;1). In thethirdchapter,weintroducetheSz asz-DurrmeyertypeoperatorsbasedonBoas- Buck typepolynomialswhichincludeBrenke-typepolynomials,Shefferpolynomialsand Appell polynomials.WeestablishthemomentsoftheoperatorandaVoronvskajatype asymptotic theoremandthenproceedtostudytheconvergenceoftheoperatorswiththe help ofLipschitztypespaceandweightedmodulusofcontinuity.Next,weobtainadi- rect approximationtheoremwiththeaidofunifiedDitzian-Totikmodulusofsmoothness. Furthermore, westudytheapproximationoffunctionswhosederivativesarelocallyof bounded variation. i In thefourthchapter,weobtaintherateofapproximationofthebivariateBernstein- Schurer-Stancutypeoperatorsbasedon q-integersbymeansofthemoduliofcontinuity and Lipschitzclass.WealsoestimatethedegreeofapproximationbymeansofLipschitz class functionandtherateofconvergencewiththehelpofmixedmodulusofsmooth- ness fortheGBSoperatorof q-Bernstein-Schurer-Stancutype.Furthermore,weshowthe comparisons bysomeillustrativegraphicsinMaplefortheconvergenceoftheoperators to somefunctions. In thefifthchapterwestudytheapproximationpropertiesofthebivariateextensionof q-Bernstein-Schurer-Durrmeyeroperatorsandobtainedtherateofconvergenceofthe operators withtheaidoftheLipschitzclassfunctionandthemodulusofcontinuity. Here, weestimatetherateofconvergenceoftheseoperatorsbymeansofPeetre’s K- functional. Then,theassociatedGBS(GeneralizedBooleanSum)operatorofthe q- Bernstein-Schurer-Durrmeyertypeisdefinedanddiscussed.Furthermore,weillustrate the convergencerateofthebivariateDurrmeyertypeoperatorsandtheassociateGBS operators tocertainfunctionsbynumericalexamplesandgraphsusingMaplealgorithm. In thesixthchapter,Wediscussthemixedsummationintegraltypetwodimensional q- Lupas¸-Phillips-BernsteinoperatorswhichwasfirstintroducedbyHoneySharmain2015. WeestablishaVoronovskajatypetheoremandintroducetheassociatedGBScase(Gener- alized BooleanSum)oftheseoperatorsandwestudytherateofconvergencebyutilizing the Lipschitzclassandthemixedmodulusofsmoothness.Furthermore,weshowtherate of convergenceofthebivariateoperatorsandthecorrespondingGBSoperatorsbyillus- trativegraphicsandnumericalexamplesusingMaplealgorithms. In theseventhchapter,weobtainthedegreeofapproximationfortheKantorovich-type q-Bernstein-Schurer operatorsintermsofthepartialmoduliofcontinuityandthePee- tre’sK-functional.Finally,weconstructtheGBS(GeneralizedBooleanSum)operators of bivariate q-Bernstein-Schurer-Kantorovichtypeandestimatetherateofconvergence for theseoperatorswiththehelpofmixedmodulusofsmoothness. In thelastchapter,weestablishtheapproximationpropertiesofthebivariateoperators which arethecombinationofBernstein-ChlodowskyoperatorsandtheSz´asz operators ii involvingAppellpolynomials.Weinvestigatethedegreeofapproximationoftheopera- tors withthehelpofcompletemodulusofcontinuityandthepartialmoduliofcontinuity. In thelastsectionofthepaper,weintroducetheGeneralizedBooleanSum(GBS)of these bivariateChlodowsky-Szasz-Appelltypeoperatorsandexaminetheorderofap- proximation intheB¨ogel spaceofcontinuousfunctionsbymeansofmixedmodulusof smoothness.