The Study Of Some Approximation Problems In Orlicz Spaces

In 1859, Cheyshev presents the characteristics of the best approximation theorem, in 1885, Weierstrass established the famous theorem that the continuous functions, can be uniform approximated by certain polynomials. Since then, the function approximation theory of this discipline was born. Later, through the study of Jackson, Bernstein, as well as a series of in-depth research of the other mathematicians, it promoted the function approximation theory to flourish. There are quite a number of scholars engaged in work in this field, and in the continuous function space and Lp spaces linear operator, interpolation polynomial, rational function approximation, etc., as a research tool issue has been a lot of results. As more and more complex problems and nonlinear problems arise, in a broader function space to study the issue has become an inevitable choice. Since Orlicz space than continuous function space and Lp space covering more extensive, its topology is also more complicated than Lp space, so in Orlicz space research function approximation problem has a certain sense of expansion.The author according to previous ideas, methods, in Orlicz space discussed some operators, interpolation, rational function approximation and some estimates width,and obtain the corresponding approximation properties. This thesis has five chapters.Chapter 1 introduces some knowledge of Orlicz spaces and related concepts and basic properties of n-width.Chapter 2 considers the approximation of linear operators in Orlicz spaces, consisting of two parts. Part 1 studies the Lipschitz properties of a class of generalized Sikkema-Kantorovich operators by using Jensen theorem. Part 2 using commonly used tools and methods of the function approximation theory, through talking about the norm of integral quasi-Kantorovich-Bezier operators, two kinds of degree of approximation of this operators in Orlicz space are obtained.Chapter 3 researches the problem of rational approximation in Orlicz spaces, considering of two parts. Part 1 studies the speed estimation of Muntz rational approximation for smoothness functions by using the Bak operator. To improve the approximation rate of smoothness functions, we introduce a new type of generalized class of Muntz rational functions and the rate of approximation is better than the usual. Part 2 studies the Muntz rational approximation of two kinds of special function classes by using the method of construction and the skill of inequalities, and gives the corresponding estimates of approximation rates of these classes under widely conditions.Chapter 4 considers the approximation of a kind of Hermite interpolation for anti periodic functions in Orlicz space, the result is better than the previous result. Then we study the approximation of a class of modified of quasi Hermite-Fejer trigonometric interpolation polynomial in Orlicz space, the estimation of approximation order is obtained.Chapter 5 studies n-width problems, consisting of two parts. Part 1 studies n-K width of certain function classes defined by linear operators in L2 space, and obtained the asymptotic estimates of n-K width. Part 2 studies the optimal approximation degree of the sample stripe space approach periodic differentiable function by using the best approximation of the duality principle, Holder inequality and other tools, with abstract approximation methods and techniques, gives the best approximation degree estimation.