Approximation Property On Generalized Bernstein Operators

Durrmeyer-type operators is an important branch of approximation theory.It's fundamental researches are similar to other positive linear operators,including the approximation problems such as approximation properties,positive and negative theorems and shape preserving properties.Based on the predecessors,we discuss the related approximation problems on the(p,q)-Durrmeyer-Schurer operators of one variable and two variables and the(p,q)-Durrmeyer-Stancu operators,which are mainly divided into the following sections:In the first chapter,we first introduce the development process and research background of approximation theory.Then we introduce the research results about Durrmeyertype operators,q-type operators and(p,q)-type operators at home and abroad in detail.Finally,we briefly describe basic definitions,theorems and computational notations used in this article and outline the structure of the main contents.In the second chapter,we define the(p,q)-Durrmeyer-Schurer operators by generalizing the Durrmeyer-Schurer operators and further study the related approximation properties by combining with K-functional and modulus of smoothness,which extends the application of Durrmeyer-type operators in(p,q)-type operators.In the third chapter,we introduce the biviriate(p,q)-Durrmeyer-Schurer operators based on the(p,q)-Durrmeyer-Schurer operators introduced in the second chapter.Then we combine with the definitions and properties of modulus of smoothness of the binary operators and discuss the convergence and related approximation results,which extends the application of binary operators in(p,q)-type operators.In the fourth chapter,we first construct the(p,q)-Durrmeyer-Stancu operators based on q-Durrmeyer-Stancu operators.Secondly,we study the approximation problems such as the convergence rate and the Voronovskaja type theorem of the operators and show figures for the operators approximating to a specific function.Finally,we establish the modified(p,q)-Durrmeyer-Stancu operators and get a better approximation result.In the fifth chapter,we mainly discuss the rational approximation to |x|?(1 ? ? < 2)at the Newman nodes and obtain the asymptotic formula by positive inequality and inverse inequality.And the approximation of order can not be improved.In the sixth chapter,we review and summarize this paper and make a prospect for further researches on Durrmeyer operators as well as(p,q)-type operators.