A Study On Some Operators Approximation Problems In Orlicz Spaces

Dating back to its source, the approximation theory of function began in 1885 when German mathematician Weierstrass established the theorems which about continuous functions can be approximated by certain polynomials and in 1859,the best approximation theorem was proposed by Soviet mathematician Chebyshev.The establishment of these two theorems made approximation theory of function become an important part of modern Mathematics.In recent decades, a growing number of scholars and research had been finished in the continuous space and Lp space. In this paper,the author will rely on the existing research and work out the approximation problems in a broader Orlicz space.It has always been a hot and difficult problem to research the appro-ximation of operators'inverse theory. In this article,the author made some approaches for the approximation problems of linear operators in Orlicz spaces, thinking about the positive theorem and the inverse theorem of the approximation of operators,similarly, a strong inverse theorem of the approximation of operators was given.The first chapter introduces the relevant knowledge and related symbols about Orlicz space.In the second chapter, study the linear operator approximation problems in Orlicz spaces, obtain the corresponding Approximation Theorem.It is divided into four parts:The first part, according to the nature of the Bernstein-Kantorovich operators to K-functional and continuous mode as a tool to study the Bernstein-Kantorovich-Sikkema-Bezier operator approximation in Orlicz space.The second part, according to the nature of the Baskakov-Durrmeyer operators,use smooth mold,Hardy-Littlewood maximal function research relevant conclusions with the Baskakov-Durrmeyer-Bezier operators approximation in Orlicz space.The third part, Agrawal and Thamer gives a definition to a new class of positive linear operators,This dissertation discusses about the nature of its approach within the Orlicz Space on the basis of analyzing the following three concepts modulus of smoothness, N-function and Jensen inequality,showing us proof of the approximation Direct and Inverse Theorems.The fourth part, in Agrawal and Thamer defined a new class of positive linear operator, and a basis for discussion on the operator while unbounded function approximation problem, continue discussion of the operator approximation in Orlicz space, showing us proof of the approximation strong type inverse theorem.