ON CERTAIN LINEAR METHODS OF APPROXIMATION

Abstract

The present thesis deals with the approximation properties of some well-known operators and their new generalizations which include the Stancu type generalization and q-variant of the well known operators. We divide the thesis into seven chapters. In chapter 0, we mention historical background, some notations and basic de nitions of approximation methods which will be used throughout the thesis. In chapter 1, we study the direct theorems, which include the point-wise convergence and the Voronovskaja kind asymptotic formula for Baskakov-Durrmeyer-Stancu operators in both ordinary as well as in simultaneous approximation. Also, the better error estimation has been obtained by using King's approach. Finally, the estimate of error in terms of higher order modulus of continuity is established in simultaneous approximation. We use the technique of linear approximating method viz. Steklov means to prove the error estimation. In the chapter 2, we study the rate of convergence for generalized form of certain summation-integral type operators, (namely Srivastava-Gupta operators) for functions having the derivatives of bounded variation. We also present the results in the simultaneous approximation. The aim of the next chapter is to introduce and study some approximation properties i.e. star-shapedness and monotonicity for discrete q-Beta operators. We establish some global direct error estimates for the above operators using the second order Ditzian-Totik modulus of smoothness. In the chapter 4, we proposed the q-analogues of well-known modi ed Beta operators and Baskakov-Durrmeyer-Stancu operators. These q-analogues of such operators are based on the q-Beta functions of the second kind. We obtain direct local approximation theorems for the operators the q-modi ed Beta operators using the i second order modulus of smoothness. Also, the similar results for the q-Baskakov- Durrmeyer-Stancu operators are established. Finally, the better error estimation has been obtained by using King's approach. In the chapter 5, we deal with the Stancu type generalization of the complex Baskakov operators in compact disks, which provides the overconvergence from real to complex domain. Actually, the complex Baskakov-Stancu operators V ; n can be written in the form of divided di erence. We obtain the upper bound, a Voronovskaja type result with quantitative estimates for these operators attached to analytic functions on compact disks. We also estimate the exact order of approximation for the operators V ; n : In the last chapter, we study the approximation properties of the complex Favard- Sz asz-Mirakjan-Stancu operators, we obtain an estimate of error in the approximation of f by S ; n . A Voronovskaja type result with quantitative estimates for these operators attached to analytic functions on compact disks is discussed. The exact order of approximation in ordinary and simultaneous approximation for S ; n are also obtained.