Some Approximation Properties Of The Gamma Type Operators

Some approximation properties of the Gamma operators and the new Gamma type operators are concerned , including the rates of convergence for functions, which are continue ,or have only discontinuity points of the first kind. The author investigates simultaneous approximation by Gama operators and the rate of pointwise convergence by new Gamma type operators.In Section one, the rencent advance in the study of simultaneous approximation pointwise approximation, and the equivalent theorems by linear positive operators. Some fundamental concepts and definitions involved in the thesis, as well as the work we do on problems are introduced.In Section two,《On the Simultaneous Approximation of Functions and Their Derivatives by the Szasz-Mirakyan Operator》which was published in《Journal Approximation Theory》states simultaneous approximation by Szasz-Mirakyan operators for functions which derivatives have only discontinuinty points of the first kind. Qi Qiulan and Guo shunsheng are interesting in this problem and point out that Bernstein-Type operators and Szasz-Type operators have the analogy results in《Simultaneous Approx- imation by Baskakov-Durrmeyer Operator》. Post-Widder operators contitute the real inversion formula for the Laplace transform. Using analysis and some inequalities, the author obtains the degree of approximation by post-widder operators for functions which derivatives have only discontinuinty points of the first kind on [0, +∞) and which are power growth.In Section three, the equivalent theorems on Gamma operators have been investigated by many people and the results become more consummate with the developing of the moduli of smoothness. In the spirit of Zhou Xinlong's work in《About Bernstein operators》, the author gives the equivalent theorems on simultaneous approximation for the combinations of Gamma operators by the weighted modulus of smoothnessωφλr(f,t)ωφs(0≤λ≤1). The author also gives the relation between the derivatives of the combinations of Gamma operators and the smoothness of derivatives of functions.In Section four, Karsli introduces a new kind of Gamma operators and obtains an estimate of the rate of pointwise convergence of these new kind Gamma operators for functions with derivatives of boundde variation. The author investigates the behavior for the new Gamma type operators and estimates the rate of pointwise convergence for the locally bounded functions.