8 - Existence and Characterization of Best Approximations

Summary

In this chapter we study the question of the existence and characterization of best approximations from the space of linear combinations of ridge functions with a finite number of directions. That is, from the space given by the restriction of

to a domain K ⊆ Rⁿ, and to where all the f_i(Aⁱx) lie in an appropriate normed linear space. These normed linear spaces will be X = L^p(K), p ∊ (1, ∞), and X = C(K).

Section 8.1 contains some general results regarding existence and characterization of a best approximation from a linear subspace. In Section 8.2 we consider the space L^p(K) for p ∊ (1, ∞), and highlight the case p = 2, while Section 8.3 contains a few simple examples of that theory. In Section 8.4 we look at C(K) where, unfortunately, we only have results when approximating from linear combination of ridge functions with two directions. Very little seems to be known about these questions in other normed linear spaces.

General Results

In approximation theory, a set M in a normed linear space X is said to be an existence set (sometimes called a proximinal set) if to each G ∊ X there exists at least one best approximation to G from M. That is, to each G ∊ X there exists an F* ∊ M satisfying

A necessary condition for M to be an existence set is that it be closed. But closure, in general, is insufficient to guarantee existence.

Closed convex subsets of finite-dimensional subspaces are existence sets. But linear combinations of ridge functions of the form (8.1) are not finite-dimensional, unless K is very restricted. However, closed convex sets of a uniformly convex Banach space are existence sets.