Of Bernstein-type Operator Column And Its Integral Deformation Completely Asymptotic Expansion

In this paper, we mainly discuss the Complete Asymptotic expansion of the Bernstein-type operators and their intgral modifications. Here, the Bernstein-type operators include the Bernstein operators, the Szász-Mirkjan operators, the Baskakov operators and the Meyer-Konig and Zeller operators. They have two kinds of integral modifications. One is so called the Kantorovich modifications, the other is the so called the Durrmeyer modifications. And the Complete Asymptotic expansion of the Bernstein operators and the Meyer-Konig and Zeller and their two kinds integral modifications have been given. Besides, the Complete Asymptotic expansion of the Baskakov operators' Durrmeyer modifications is also given. In this paper, we get the Complete Asymptotic expansion of otheir two operators and their two kinds modifications.In fact, the Asymptotic expansion of the operators is operators approximating to functions in higher orders. It characterize the order of approximation by the operators, on the other hand, it also is an preparation of the investgation of the inverse theory of approximation. Asympototic of operators has many kinds, such as pointwise, uniform norm, p norm and so on. In this paper, we will give Complete Asymptotic expansion in the sense of pointwise.In the first section, we first introduce the development of the Bernstein-type operators and the Asymptotic expansion of them and their integral modifications, then, we introduce the knowledge and conclusions which we need in this paper. In the same time, we will found two important equtaions in combinatorics.In the second section, we will get the Complete Asymptotic expansion of operators V_n(f; x) base on the relations of the operators V_n(f; x) and M_n(f; x), we also get the the Complete Asymptotic expansion of the Kantorovich modifications of the operators V_n(f;x). Of course, these include the Voronovskaja-type Asymptoticexpansion of them. In the third section, we get the Complete Asymptotic expansion of operators S_n(f; x) and their two kinds integral modifications base on the two key equations which we get in the first section (Lemma 1.3, Lemma 1.4) and the ordinary theorem of approximation expansion (Theorem A), as their corollary, we also get the Voronovskaja-type Asymptotic expansion of them.The stduy we did cause the Complete Asymptotic expansion of the Bernstein-type operators and their intgral modifications in the sense of pointwise have a relativly complete results. The study we did also illustrate that the close relations of the different branches of mathimatics.