The Sun Set In Orlicz Spaces

The proplem of nonlinear approximation have been studied since the end of the eighteenth century,The mathematician P.L.Chebyshev posed and discussed uniform approximation of the rational functions,but when dealing with the problems,they always put the focus on polynomial approximation,the essential studies about the problems of nonlinear approximation have been broken since the last century.The investigations of approximation theory have last for a long time,the type of its development obey the regular from concretion to generalization , this thesis will discuss the problems from generalization to concretion.The sun set in Orlicz spaces and its approximation properties are investiga ted in this thesis.Moreover, the geometric properties in the whole spaces served as simple corollaries will be given.The paper consists of three parts:The main results of the thesis are summarized as following:Chapter 1 Introduction:It is reviewed that developing course of the theory and geometric theory of Orlicz spaces during more than 60 years studies.Moreover,it is evaluating and summarizing the pioneer's main research results,and showing the background and significance of the content of each part in this thesis.Chapter 2 The sun set in Orlicz sequence spaces:Discuss the sun set in Orlicz sequence spaces and give necessary and sufficient condition of several theorems for that a set equipped with Orlicz norm is a sun set in Orlicz sequence spaces.The kind of disscussment has been verified in general Banach spaces,moreover,the necessary and sufficient conditions that a set endowed with Luxemburg norm is a sun set in Orlicz sequence spaces have been given.The Chapter will go step further on this base.Chapter 3 The approximation properties of the sun set in Orlicz function spaces:On the base of the results that the characteristic theorems that general subset G and subset LP (T ,Y)(Y ? X)of particular type in LP (T ,X) are sun sets,The...