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Approximation theory in normed spaces

Stavroulakis Dimitrios

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Summary

Approximation Theory is a branch of Mathematical and Numerical Analysis, depending on the "point of view" from which we will examine it. Its theoretical aspect focuses on the existence of an optimal approximation and its uniqueness, as well as on theoretical issues in which they find applications. On the other hand, its practical one involves numerical methods, error estimates and other tools that help to find an approximation and convert it into an optimal one.The purpose of this paper is to examine mainly the theoretical aspect of approximation theory in normed spaces. Firstly, we present the necessary introductory concepts and definitions, investigate the special cases of normed spaces (Banach spaces, Hilbert spaces) and restrict ourselves to finite-dimensional spaces in order to formulate and prove the fundamental theorem of approximation theory, which ensures the existence of an optimal approximation for elements (vectors, functions) of the above spaces.Secondly, we give the definition of strict convexity, prove the existence of at most one optimal approximation for elements of the space if it is strictly convex, and the existence of infinite optimal approximations for some element of the space in the opposite case. The remainder of the chapter is devoted to the problem of uniform polynomial approximation of continuous functions. Its existence is guaranteed by the 1st Weierstrass approximation theorem for which the proofs of Landau & Bernstein are given. Besides, we consider the conditions under which the uniform polynomial approximation becomes optimal and its uniqueness is ensured.In the last part of the thesis, we discuss the existence of nearest points in Banach spaces with the Radon-Nikodym property. The definitions of nearest points and weak proximinality are given first, followed by the characterization of Banach spaces having the Radon-Nikodym property, via Bochner integrable functions. We then consider the relation between the Radon-Nikodym property and that of dentable sets and conclude with the Borwein & Fitzpatrick theorem, which ensures the existence of a nearest point in a non-empty, closed and bounded subset of Banach space. Finally, we state Edelstein's theorem, which gives an additional property to the set of points with a nearest point to the aforementioned subset of space.

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