Two Classes Of Approximable Functions And Their Applications

Function approximation problem has always been an important research topic in mathematics. Since 1920 s, a lot of important results had been obtained. At the same time, a lot of famous problems, such as Bernstein’s Approximation Problem, had been raised. Until today, these results and theorems have attracted much attention. Many new theorems and methods have also been arised. This thesis deals with two classes of approximable functions. One can be rapidly approximated by polynomials. We call it rapidly approximable functions(by polynomials). The other one can be approximated by periodic functions when the variable is su?cient large. This class includes asymptoticallyω-periodic functions and ω-periodic limit functions.Firstly, we investigate some problems about rapidly approximable functions. Zerner and Zeriahi have already described the space of rapidly approximable functions for a bounded set respectively. However, how to describe the space of rapidly approximable functions for a unbounded set is still open. We discuss this question with the weights Wα =e-|x|α(α > 1), which form a subclass of Freud weights on the real line. We investigate the subspace of L2(R, W2αdx) consisting of those elements that can be rapidly approximated by polynomials. This subspace has a natural Fr′echet topology, in which it is isomorphic to the space of rapidly decreasing sequences. We show that it consists of smooth functions and obtain concrete results on its topology. For α = 2, there is a complete and elementary description of this topological vector space in terms of the Schwartz functions.Secondly, we focus on how to characterize the asymptotically ω-periodicity. Some necessary and su?cient conditions of asymptotically ω-periodic functions are given. Asymptotically ω-periodic functions in the Stepanov sense are also considered. These functions extend the one of asymptotic ω-periodic functions to include locally integrable functions.Some necessary and su?cient conditions of asymptotically ω-periodic functions in the Stepanov sense are given. Then we use necessary and su?cient conditions of asymptotically ω-periodic functions(in the Stepanov sense) to investigate the existence and uniqueness of asymptotically ω-periodic mild solutions to a class of semilinear fractional integro-di?erential equations.Thirdly, we definite a new class of functions what we call ω-periodic limit functions. These functions can be approximated by measurable periodic functions pointwise when the variable is su?cient large. We show the new function space is a Banach space and some other properties. In particular, we study inclusion relations among asymptotically periodic type function spaces. Finally, we apply the ω-periodic limit functions to investigate the existence and uniqueness of asymptotically ω-periodic mild solutions of an abstract Cauchy problem.