The Research About Q-Durrmeryer Operator’s Approximation Property

The idea of approximation is widely used in many disciplines. Many researchers have studiedsome problems in the approximation theory, see[1 4].Xie Tingfan and ZhouSongping[1]investigated the polynomial approximation, Fourier approximation, operatorapproximation, interpolation approximation etc. and relative problems. Wang Renhong has studiedsome problems in function approximation,see[43 45].Z.Ditzian and V.Totic[2]studied theK-functional and moduli of smoothness in detail. G. M. Phillips[3]firstly introduced theq-Bernstein operator by virtue of the q-integer, which extended the study of the Bernstein operators.Vijay Gupta[4]introduced the q-Durrmeyer operator by the q-integer which was the extension ofthe Bernstein- -Durrmeyer operator. Vijay Gupta and Wang Heping[5]introduced another type ofq-Durrmeyer operators, which are better than the operators in[4]. brahim Büyükyazici[6]presented a novel class of Bernstein-type operators.The organization of this dissertation is as follows: In Chapter 1，we introduce some necessarydefinitions and notations. In Chapter 2, we first give some approximation properties of q-Durrmeyeroperator on some class of functions by virtue of properties of q-Durrmeyer operator and theH lder’s inequality[17]. Next, we give the recurrence formulae and some approximation propertiesof q-Durrmeyer operator. In Chapter 3, we study a novel class of q-Durrmeyer operators and givetheir some approximation properties. In Chapter 4, we investigate some approximation properties ofq-Durrmeyer operators in the Orlicz space.