A Research Of Approximate Properties Of Beta And Gamma Type Operators

This paper is studied for the approximation of Beta operatorsβ_n(f, x) for absolutely continuous functions and bounded funtions, and the global approximation theorem for Gamma operators G_n(f, x) in L_p. This paper is organized as following:In chapter 1 we give an explanation of definitions and marks which were appeared in the paper and then introduce some researchful results.In chapter 2 we introduce a class of Beta operatorsβ_n(f,x), we study the rate of convergence of Beta type operators for some absolutely continuous functions and bounded functions, using classical decomposed method of Bojanic—Cheng and analytical ways, we estimate the first order absolute momentβ_n(|t-x|, x) and sigh functionsβ_n(sign(t-x), x) respectively, we get a better result of the rate of approximation for |β_n(f,x) - f(x)| and asymptotic formula.In chapter 3 The approximation properties of the Gamma operators in the spaces L_p(0,1) are discussed. We use the technique of interpolation spaces and characterization of K—functional and modulus of smoothness which has been used for inverse theorems. The equivalence theorem concerning the global aproximation by the Gamma operators is established.