Asymptotic Expansions Of Probabilistic Type Operators

In the present paper we explicit how some well-known Bernstein-type operators converge to Szasz operators in an approximate asymptotic sense. As a main tool, asymptotic expansion is expressed by using semigroup of linear operators. This paper is organized as following:In chapter 1 we give an explanation of definitions and marks which are appeared in the paper and then introduce some researchful results, we introduce the conception of semigroup of operators, and give the presentation of linear operators by semigroup of operators.In chapter 2 we give some lemmas which are useful in the whole context.In chapter 3 we obtain the asymptotic expansion theorems of Bernstein operators, Bleimann, Butzer and Hahn(BBH) operators and Baskakov operators.In chapter 4 the corresponding asymptotic expansion theorems of Bernstein-Kantorovich operators and Baskakov-Kantorovich operators are obtained.In chapter 5 we show that the methods of semigroup of operators can be easily extended to multivariate linear operators, we chose the bivariate Bernstein operators as an example to reveal this assertation.