Estimates Of Voronovskaja Type Of Bernstein Operator And Its Generalizations

Because of its simplicity in construction,and its excellent properties in keep-ing monotonicity and convexity of the objective functions,Bernstein operator has played very important roles in operator approximation and even in the w-hole function approximation theory.Bernstein operators have been widely used in functional analysis,computational mathematics and other fields.The main purpose of the present thesis is to consider the Voronovskaja type estimations of Bernstein operator and its important generalizations.The main results can be read as follows:Chapter ?.We give a brief survey on the research of Bernstein operator and some important generalizations of it.Chapter ?.Gonska and Tachev[5]obtained a kind of the Voronovskaja type estimate of Bernstein operator.The main purpose of this chapter is to generalize their conclusions to the weighted case.In fact,We obtain a weighted Voronovskaja type estimate of the Bernstein operator with Jacobi weight w(x)= xa(1-x)bChapter ?.Let Cw= {f(x)?C[0,1]:limx?1(wf)(x)=limx?0(wf)(x)= 0},where w(x)= xa(1-x)b is the Jacobi weight.As we know,the functions in Cw may have singularities at the endpoints.By using the weighted K-functional and the weighted modulus of smoothness,we obtain Voronovskaja type estimate for weighted approximation by a kind of modified Bernstein operators Bn*(f,x)in Cw.Chapter?.Gadjiev and Ghorbanalizaeh[21]introduced a new Bernstein-Stancu type operator Sn,?,?(f,x),and obtained some approximation properties of the new operators.In[21],it is showed that Sn,?,?(f,x)can be applied to approximation the continuous functions defined An(a subset of[0,1]).Wang,Yu and Zhou[22]discovered that Sn,?,?(f,x)actually can approximate the functions in C[0,1]as well.Our aim of this chapter is to study the Voronovskaja type estimate of Gadjiev's operator on the An.Chapter ?.We study the approximation properties of a new Kantorovich-type Bernstein-Stancu operator introduced by Icoz[241 on the[0,1]interval,and obtain its Voronovskaja type estimate.