The Study Of Some Weighted Approximation Problems In Orlicz Spaces

The approximation theory of function studied about using simple and computable functions to approximate general functions, thinking about the approximation degree and how to descript the characteristic of the approximated function. The approximation theory of function mainly studied about these problems, such as, approximation by linear operators, interpolation approximation, rational approximation,algebraic polynomial approximation, trigonometric polynomial approximation and so on. Most work were stuied in continue space and Lp space, however, it was relative deprivation in Orlicz space. So, it was great meaningful to study the approximation degree in Orlicz space.There were four chapters:Preliminary knowledge?approximation by linear operators in Orlicz space?Interpolation approximation in Orlicz space?Muntz rational approximation in Orlicz space.We introduced some relative knowdge and marks in Chapter oneApproximation by linear operators in Orlicz space was studied in chapter two which divided into two parts:in the first part, we considered the convergence and boundedness of a kind of generalized Baskakov-Kantorovich Operators with one and two variables in Orlicz space. Under the modules of smoothness, the great function of Hardy-Littlewood, Steklov average function and the Jensen inequality the estimation of degree of weighted sense in Orlicz space were given. In the second part, the properties of modified Meyer-Konig-Zeller operators were studied in orlicz space, Ditzian-Totick-modulus,new definition of K-function, Holder inequality were used as tools to work out the direct and inverse theorems of the operators in Orlicz space.Interpolation approximation in Orlicz space was studied in chapter three. In the first part, we modified appropriately the Shepard-Lagrange interpolation operators, the properties of modified Shepard-Lagrange type interpolation operators were studied in Orlicz space, the boundedness of the interpolation operators were given, the convex property of N function, Jensen inequality, Hardy-Littlewood great function were used as tools to estimate the approximation of the interpolation operators in Orlicz space. In the second part, we popularized the second order Hermite interpolation operators, the properties of the third order Hermite interpolation operators were studied in Orlicz space.Muntz rational approximation in Orlicz space was studied in chapter four. In the first part, studied on the properties of weighted rational function of Muntz in Orlicz space, used the weighted modulus of continuity, K-function, Hardy-Littlewood great function and Holder inequality as tools to estimate the approximation of the rational function in Orlicz space. In the second part, we introduced a new kind of modified Kantorovich-Bak operators. The approximation degree was studied in weighted Orlicz space and we also found the Jackson type theorem.