Convergence estimates for linear Positive operators

Abstract

newlineABSTRACT newline newlineThe thesis entitled CONVERGENCE ESTIMATES FOR LINEAR POS- newlineITIVE OPERATORS is a part of research work done by me primarily fo- newlinecused on the investigation of approximation properties of some well known newline newlinelinear positive operators and their new generalizations, such as, modified newlineBaskakov-Durrmeyer operators, Bivariate Sz and#769;asz-Durrmeyer operators, and newlineBivariate extension of and#955;-hybrid operators. newline newlineWe divide the thesis into five chapters: newline newlineThe Chapter 1 is of introductory nature it contains literature survey, pre- newlineliminary definitions and notations of approximation methods which will be newline newlineused throughout the thesis. We also discussed some basics tools to measure newlinethe quantitative approximation of functions by linear positive operators, e. newlineg., the modulus of smoothness of first and second order, Peetre K-functional newlineand Ditzian-Totik modulus of smoothness . newline newlineThe Chapter 2 deals with approximation properties by modified Baskakov- newlineDurrmeyer operators. We construct a new sequence of modified Baskakov- newlineDurrmeyer operators including the shape parameter alpha and study the newline newlineuniform convergence of these operators by means of modulus of continuity newline newlineto the continuous functions. We investigate the point wise and weighted ap- newlineproximation in terms of Ditzian-Totik uniform convergence with the aid of newline newlinefirst and second order of modulus of smoothness. Further, we calculate the newlinedirect estimate of rate of convergence in terms of Lipschitz-function. Lastly, newlinewe study weighted approximation result. newline newlineThe Chapter 3 devoted to construct a new sequence of bivariate Sz and#769;asz- newlinevii newline newlineviii ABSTRACT newlineDurrmeyer operators based on Dunkl analogue. We investigate the order of newlineapproximation with the aid of modulus of continuity in terms of well known newlinePeetre s K-functional, weighted approximation results. Voronovskaja type newlinetheorems and Lipschitz maximal functions. Further, we also discuss here the newline newlineapproximation properties of the operators in B and#776;ogel-spaces by use of mixed- newlinemodulus of continuity. newline newlineIn Chapter 4, we proposed to construct a new family of Hybrid type newline newlineoperators of summation integral type via shape parameter alpha. We inves- newlinetigate basic approximation results in terms of classical Korovkin theorem and newline newlinemodulus of continuity. For our operators, we discuss local approximation re- newlinesults by means of Peetre s K-functional, second order modulus of smoothness, newline newlineLipschitz class and Lipschitz maximal function. Next, we discuss weighted newlineapproximation theorems in terms of weighted Korovkin type theorem and newline newlineweighted modulus of continuity. Lastly, A-Statistical result and rate of con- newlinevergence for functions with derivative of bounded variation are also studied. newline newlineThe Chapter 5 is concerned with the development of a bivariate exten- newlinesion of hybrid type operators. We discuss the order of approximation via newline newlinemodulus of continuity, Peetre s K-functional, the rate of convergence, Lips- newlinechitz maximal functions and Voronovskaja type result. In addition to this, newline newlinewe investigate global approximation results. In the last section, we study newlinethe approximation properties of the operators in B and#776;ogel-spaces in terms of newlinemixed-modulus of continuity.