The Preservation Problem Of Integral Deformation Positive Operators

In this paper, we mainly discuss the preservation problem of general Kantorovich-type operators and several Durrmeyer integral deformation operators of the classical Bernstein-type operators.The positive linear operators have a simple form and a good preservation and approximation property, the study of which have important theoretical significance and wide application prospect. At presently, there have been abundant in study for the kantorovich integral deformation of classical Bernstein-type operators by analytical techniques and have obtained many conclusions [Documents 3-5]. But we don't have any result about the preservation property of general Kantorovich-type operators. Here, the general Kantorovich-type operators are these operators that they can be written in the form:where L_n is a positive linear operator,φ_n(t) is nonnegative function defined on I and T_n is stretch operator: T_n = f(c_nu) , c_n is constant indepenting n. We request 0 < c_n ≤ 1 and 0 ≤ φ_n(t) ≤ 1 - t as I = [0,1].In this paper we shall consider the problem that under what condition, the operator L_n still possesses the preservation properties of L_n. In the second section , we give the properties about the operator L_n which preserves monotonicity, convexity ,starshape, subadditivity, variation-diminishing property and so on (Theorem 2.1,2.3,2.4,2.5,2.8).We found the probabilistic representation about Bernstein-Durrmeyer, Sz(?)sz-Durrmeyer and Baskakov-Durrmeyer operator, thereby we obtain their monotonicity preservation property , convexity preservation property , smoothness preservation property and diminish bounded-variation property by probabilistic method (Theorem 3.1-3.3). These are the content of the third section.The results in the thesis relatively complete the study of the preservation properties of the classical Bernstein-type operators and integral deformation of which and further the study of the preservation properties of the general Kantorovich-type operator.