Some Approximation Problems In Orlicz Spaces

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 20^th century. With the development of science and technology,the relationship between the approximation theory of functions and other applied subjects is more and more closely related. For decades of years,a large number of scholars from all over the world studied in this field and have obtained masses of scientific results in the LP spaces and the continuous functions spaces. But in other spaces which are more extensive,such as Orlicz spaces and so on,the study results are only few. Thus,this paper studied some approximation problems in Orlicz spaces,such as approximation by linear operators and reciprocals of polynomials,as well as some n-width problems. This thesis has four chapters.Chapter 1 introduces some knowledge of Orlicz spaces and related concepts and basic properties of n-width.Chapter 2 considers approximation by linear operators in Orlicz spaces, consisting of three parts. Part 1 and Part 2 study the convergence of approximation of Kantorovich-Shepard operators whenλ> 1andλ= 1 separately, obtain the degree of approximation by means of k-functional and modulus of continuity. Part 3 studies a kind of generalized Kantorovich operators and obtains the sufficient and necessary condition for approxima- tion and the estimate of the degree of approximation.Chapter 3 considers approximation by reciprocals of polynomials, consisting of two parts. Part 1 studies approximation by reciprocals with complex coefficients and obtains the degree of approximation by usingΔcondition of N-functions. Part 2 studies approximation by reciprocals of polynomials with positive coefficients and obtains the degree of approxima- tion.Chapter 4 studies n-width problems, consisting of two parts. Part 1 studies n-K width of certain function classes defined by linear differential operators A=∑a_k(x)D^k and obtained the asymptotic estimates of n-K width. Part 2 discusses the problems of n-width for the Sobolev spacesΩ_∞^r and obtains the exact values of the Kolmogorov widths, Gelfand widths and linear widths. Related optimal subspaces and optimal linear operators are given.