On Approximation By Genuine Bernstein

Approximation by operators is always one of the most important branches of function approximation theory.Among numerous operators,due to its simplicity in construction,and its excellent properties in keeping geometric properties such as monotonicity and convexity of the objective functions.Bernstein operator has played very important roles in the approximation by operators.Bernstein operator and its generalizations have been widely used in computational mathematics,the image processing and many other fields.Many scholars have carried out extensive and in-depth researches to the Bernstein operators.The main purpose of the present thesis is to consider some approximation properties of the genuine Bernstein-Durrmeyer type operators in movable interval.The main results can be read as follows:Chapter I.We give a survey on the study of Bernstein opertors,genuine Bernstein-Durrmeyer type operators and Bernstein-Stancu operators in movable interval.Chapter II.We introduce a type of genuine Bernstein-Durrmeyer operators in movable interval,which can reproduce the linear functions.The approximation rate of the new operators for continuous functions and Voronovskaja's asymptotic estimate are obtained.Chapter III.We obtain the converse results of approximation by BernsteinDurrmeyer type operators in movable interval defined in Chapter 2.It is necessary to construct an auxiliary operator which has its own independent values.The moment estimates of this operator play important roles in the proof of the converse theorem.Chapter IV.We obtain the direct and converse results,Voronovskaja's asymptotic estimate of the auxiliary operators in movable interval introduced in Chapter3.