Some Approximation Problems Under L_M~* Metric

In 1859,Chebyshev,a well-known mathematician from the Soviet Union put forward the famous theorem of characterization for best approximation. In 1885, German mathematician Weierstrass established another famous theorem that the continuous functions,can be approximated by certain polynomials. Since then,the approximation theory of functions,one of the important branches of modern mathematics,began its prosperous development with the study of lots of scholars,especially the study of Jackson,Bernstein and other Soviet mathematicians in the 20th century. With the development of science and technology, the relationship between the approximation theory of functions and other applied subjects are more and more closely related.For decades of years,a large number of scholars from all over the word studied in this field and have obtained masses of scientific results in the Lp spaces and in continuous functions spaces.But in other spaces witch are more extensive,such as Orlicz spaces and so on,the study results are only few.Thus, this paper studied some approximation by linear operators and rational approximation,reciprocal of polynomials approximation as well as interpolation approximation.This thesis has five chapters.In chapter 1 we mainly introduce knowledge of Orlicz sapace and related concepts.In chapter 2 considers the linear operator approximation, consisting of two parts. Both in Orlicz spaces using the equlence of K-funtional and Modulus of continuity studied two kinds of Bernstein-Kantorovich operators obtain some approximation theories.In chapter 3 also in Orlicz spaces considers the Muntz rational appro-ximation, obtain three results. In chapter 4 considers the approximation by reciprocals of polynomials ,it is consists of two parts. In part one we consider the approximation by reciprocals of polynomials with positive coefficient and in part two we talked reciprocals of polynomials with complex coefficient.Chapter 5 consists of two parts:In part one we studied Hermite interpolation polynomials approximation, in part two we talked the approxi-mtion by trigonometric interpolation polynomial.