Equivalent Theorems On Pointwise Simultaneous Approximation By Some Linear Operators

The operator approximation is an important branch of the approximation theory, in which the approximation by positive linear operators is a very interesting research subject. A number of mathematicians have studied it and obtained many valuable research outcomes. In the present thesis, the author carries on the study of the approximation by positive linear operators. The thesis consists of three Chapters.In Chapter one, the author introduces the developmental course of the theorems concerning the approximation by linear positive operators. Some fundamental concepts and definitions involved in the thesis, as well as a summary of the present study are also given in this chapter.In Chapter two, the author gives the equivalent theorems on simultaneous approximation for the combinations of Bernstein operators by the r-th Ditzian-Totik modulus of Smoothness ωφλr(f, t)(0 ≤ λ≤1). The relation between the derivatives of the combinations of Bernstein operators and the smoothness of the derivatives of functions approximated is also investigated in this chapter.In Chapter three, the author attempts to give out the proof of some new direct and inverse results on pointwise simultaneous approximation by the combinations of Bernstein-Kantorovich operators, using ωφλ2r(f, t)(0 ≤ λ≤1), where ωφλ2r(f,t)(0 ≤ λ≤ 1) is the Ditzian-Totik modulus of smoothness.