The Study Of Some Approximation Problems In L_M^Ba Spaces

The approximation theory of functions is one of the important branches of modern mathematics.It began in the 19th century when two famous theorems were established.On says that the continuous function can be approximated by certain polynomials which were established by Weierstrass in 1885 and the other that establishes characterization for approximation by polynomials was obtained by Chebyshev in 1859.It grew prosperous in 20thcentury and became an independent subject. People have generated a series of theories and methods in order to use the simple and computable function to approximate general function.For example,the best approximation,Fourier approximation, approximation by trigonometric polynomials, approximation by algebraic polynomials, linear operators approximation, interpolation approximation, rational approximation, approximation by reciprocals of polynomials, Muntz approximation and so on. This article studied linear operators approximation, approximation by polynomials, interpolation approximation in L_M^Ba spaces which were composed of a series of Orlicz spaces.This article contains five chapters.Chapter1:In this section, ,we introduce the properties of the Orlicz spaces and the L_M^Ba spaces. Chapter2 : In the first part of the second chapter,we generalized the Stancu-Kantorovich operator in L_M^Ba spaces by means of the relation of Orlicz norm and L_M^Ba norm. In the second part of the second chapter,we studied the approximation of Sikkema-Kantorovich operators in L_M^Ba spaces by means of the weighted modulus of continuity and K-function.Chapter3 : In the first part of the third chapter,we studied the approximation of polynomials in L_M^Ba spaces . In the second part of the third chapter, we studied the two linear and monotone polynomials operators have bounds in L_M^Ba spaces by means of the equivalence of K-function and modulus of continuity.Chapter4 : we studied the approximation of Fejer-Korovkin operator, in L_M^Ba spaces by means of K-function and modulus of continuity.Chapter5 : In this section, we studied the degree of convergence two kinds of generalized Grunwald interpolation operators based the zeros of T_n ( x ) = cos n arccosx is consided in L_M,ω^Ba spaces.