Approximation Problems Of Bounded Bernstein Operators

In the research of approximation theory,the approximation problem of the positive linear operator has been deeply concerned by many scholars.In recent two years,scholars applied(p,q)-integers into approximation theory,so that appearance of a large number of(p,q)operators.In this paper,a new kind of(p,q)-Kantorovich operators is defined based on the related theories and the approximation properties of such operators are discussed.The main contents are as follows:The first chapter is the introduction part.Firstly,the development of the approximation theory and its research background are briefly described.Secondly,the development of Bernstein operators,Kantorovich operators,binary operators and its deformation operators are introduced.Finally,the definitions,symbolic descriptions and some classical approximation theorems used in this paper are introduced.In the second chapter,we generalize the(p,q)-Bernstein-Schurer-Kantorovich operator based on the(p,q)-Bernstein-Kantorovich operator.We use K-functional,smoothing modes and other tools to obtain the convergence rate and approximation theorem of the operator.Finally,we apply the King theorem to optimize the operator so that the speed of the approximation is better.In the third chapter,we define the bivariate(p,q)-Bernstein-Schurer-Kantorovich operator based on the(p,q)-Bernstein-Kantorovich operator.What is more,we compute the central moments of two variables and prove its approximation properties.In the fourth chapter,we remove the condition that f is a non-decreasing function in(p,q)-Sz(?)sz-Mirakjan-Kantorovich operator proposed by Sharma H and Gupta C,and reconstruct a new(p,q)-Szasz-Mirakjan-Kantorovich operator.Then we calculate the central moments of each order of the operator and obtain the approximation theorem,weighted approximation theorem and Voronovskaja type theorem.Finally,we modify the operator by King's theorem and make the approximation better.In the fifth chapter,we discuss the rational approximation to |x| at the adjusted tangent nodes.The approximation order is O(1/n2),and it is proved that the order of approximation can not be improved.In the sixth chapter,we summarize the whole text and make a prospect for the approximation of positive linear operators.