Some Approximation Properties Of ?-Bernstein Operators

Stancu type operator is an important research direction of function approximation theory.In this thesis,we take the bivariate ?-Bernstein-Stancu operator as the object,and study some approximation properties of this operator and its GBS operator.The full text includes five chapters,which are as follows:In chapter 1,the research history and current situation of Stancu type operator and ?-Bernstein operator are summarized,and then we give the research content and research ideas of this thesis.In chapter 2,the definition of ?-Bernstein operator and its related approximation theorem are given,besides,we extend it to the bivariate case.Then we study the approximation properties of bivariate ?-Bernstein operator and its GBS operator.In chapter 3,the ?-Bernstein-Stancu operator is obtained by studying the Stancu generalization of ?-Bernstein operator.We obtain the recursive relation and convergence rate of moments by using the moduli of continuity,and present the asymptotic results of Voronovskaya type and Grüss-voronovskaya type of the operator.In chapter 4,we study the bivariate case of the ?-Bernstein-Stancu operator.The Voronoskaya type theorem is given,and some approximation properties of the bivariate operator are introduced by combining the mixed moduli of continuity,the Lipschitz class of the bivariate function and the K-functional.In addition,the GBS operator of bivariate ?-Bernstein-Stancu operator is constructed,and the approximation of Bcontinuous function and B-differentiable function is studied.In chapter 5,the whole thesis is summarized.We look forward to the next research work.