Published online by Cambridge University Press: 05 August 2015
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 fi(Aix) lie in an appropriate normed linear space. These normed linear spaces will be X = Lp(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 Lp(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.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.