On The Rate Of Approximation By Bernstein-Stancu Operators

Since its simplicity in construction,and its good properties to keep the mono-tonicity and convexity of the objective functions,Bernstein operator has always been one of most important parts of theory of approximation by operators.Bern-stein operator has been applied widely in Functional Analysis,Computational Mathematics and Learning Theory,etc.The main purpose of the present the-sis is to consider some approximation properties of Bernstein-Stancu operator,which is an important generalization of Bernstein operators.The present thesis is organized as follows:Chapter ?.We give a brief survey on the approximation by Bernstein operator and its generalizations,for continuous functions and function with singularities.Chapter ?.In this chapter,we consider the weighted approximation of Bernstein-Stancu operator for functions with singularities.In fact,we establish both the direct and inverse theorems for weighted approximation.Our results generalize those of Della and Vecchia([45]).Chapter ?.We introduce a kind of Durrmeyer type Bernstein-Stancu oper-ator,and study its approximation properties in C[0,1],Establish both the direct and inverse theorems of approximation.Our results are pointwise,and generalize those of Dong and Yu([7])from An(C[0,1])to the whole interval[0,1].