The Study Of Some Approximation Problems In Orlicz Spaces

The studies on approximation of linear operaters,rational approximiation and interpolation approximiation are relatively perfect in continuity function spaces and Lp spaces. While in Orlicz spaces which are general spaces of Lp (P>1), the studies of these contents are slow.Therefore,in this paper, the approximation of linear operators,rational approximation ,interpolation approximation are discussed in Orlicz spaces.and obtain corresponding approximation theorems.This paper contains three chaptersChapter one the theory on approximation of linear operators,the chapter which contains three sections mainly introduce the approximation of three difference types linear operators,and obtain corresponding approximation theorems,section one with the K-functional as a tool ,we prove the direct and the inverse theorems of approximation by Bernstein-Durrmeyer operators in Orlicz spaces. Especially, an approximation equivalent theorem of the operators is also obtained . Section two with the K-functional and modulus of smoothness as the tool, we discuss Durrmeyer- Be′z ier operators in Orlicz, and an approximation direct theory is obtained . Section three, we research the linear combinations of Bernstein-Kantorovich operators and the Jackon type estimates of degree of approximiation are obtained.Chapter two rational approximation , the Chapter which contains two sections mainly study the approximation of reciprocals of polynomials and Muntz rational approximation .Section one with 2-th degree modulus of smoothness as a tool ,we discuss approximation of differentiable periodical functions by reciprocals of Trigonometric Polynomials in Orlicz spaces ,and obtain a corresponding approximation theorems .Section two,with Bak operators as a tool , we study Muntz rational approximation and prove corresponding approximation theorems.Chapter three Interplation approximation ,the chapter which contains two sections , mainly introduce the question of interplation approximation and obtain corresponding approximation theorems . Section one , we study approximation of the frist modified rational interpolation in the weighted Orlicz spaces L_ω~M . and obtain the flower estimate. Section two we study approximation of the modified Hermite- Fe′jerand lagrange interplation in L_ω~M space , and obtain the corresponding theorems.