On Approximation Properties Of Weighted Bernstein-Kantorovich Operators

In the present thesis, modified Bernstein-Kantorovich operator through weighting is introduced, which pulls approximation problems about functions into a new form. This also means that the new operator solves the approximation problems of a kind of not integrable functions. Some results about this problem are obtained. The properties of this new operator are studied, especially the formula of the standardized factor and its behavior around the two endpoints which plays a. quite important role in the approximation problem. Meanwhile, this thesis is also concerned with the q-Bernstein operator which has stirred the interest of many people home and abroad these years.Following is the structure of this thesis.The first section introduces the background and the development actuality of the Bernstein-Kantorovich operator, as well as the definition of the modified Bernstein-Kantorovich operator. The necessary declarations of the definitions and marks are also given in this part.The second section studies the properties of the new operator. The formula of the standardized factor is presented. Furthermore, the extreme properties of this standardized factor are considered, which shows the close connection between the properties and the value of x together with the essentiality of the standardized factor behavior around the two endpoints to the approximation problem. At last of this part give the maximum order of the standardized factor.The third section studies the approximation problem of a kind of not integrable functions with this new operator. First, deal with the boundness of the operator with the weighting. Then divide up [0,1] into three parts based on the properties of the standardized factor under the new norm. At last, using K-functional estimate the approximation degree in the new functions space.The forth section show the saturation theorem of q-Bernstein operator with the parameter q varying between 0 and 1 and estimate the convergence rate as the parameter q→∞through the fine calculation.