The limits of a Chebyshev-type theory of restricted range approximation (Q1103806)

Consider extended real-valued functions g(x), u(x) on \(X:=[0,1]\) subject to the restrictions: (i) \(-\infty \leq g(x)<u(x)\leq +\infty\) for all \(x\in X\); (ii) g and u are upper and lower semicontinuous on X, respectively. Let H be an n-dimensional subspace of C(X) with the Chebyshev norm and set \(K:=\{g\in H:g\leq q\leq u\}\). The problem of restricted range approximation is, given \(f\in C(X)\), to find a function \(p\in K\) such that \(\| f-p\| =\inf \{\| f-q\|:q\in K\}.\) Many authors have studied this problem and have obtained a Chebyshev-type theory including existence, alternation, characterization, and uniqueness. In this paper a detailed study is made of the limits of this theory in terms of several technical properties of the set K that the author defines and studies. A point \(p\in K\) is said to alternate k times with respect to (g,u) on X if there are \(k+1\) points \(0\leq x_ 1<x_ 2<...<x_{k+1}\leq 1\) in X such that either \(p(x_ j)=g(x_ j)\), for j odd and \(p(x_ j)=u(x_ j)\) for j even; or vice versa. K is said to be an alternation singleton if K contains only one element and this element alternates at least n times with respect to (g,u) on X. K is said to have property \(B_ 1\) if \(n>1\) implies \(K=H\), and \(n=1\) and g(x)\(\not\equiv -\infty\) (resp. \(u(x)\not\equiv +\infty)\) imply the existence of two distinct points \(x_ 1,x_ 2\) and a function \(p_ 0\in K\) satisfying \(p_ 0(x_ i)=g(x_ i)\) (resp. \(u(x_ i))\), \(i=1,2\). The set K has property C (resp. property \(\tilde C\)) if for every \(f\in C(X)\) (resp. \(f\in \tilde C(X):=\{f\in C(X):g\leq f\leq u\})\), a necessary and sufficient condition that \(p\in K\) be a best approximation to f from K is that f-p alternate at least n times on X. Some sample results: Theorem 1. K has property \(\tilde C\) if and only if K is an alternation singleton or H is a Haar subspace. Theorem 2. K has property C if and only if either K is an alternation singleton or H is a Haar subspace and K has property \(B_ 1\). The author gives also several characterizations of each of the following properties: (a) either K is an alternation singleton or H is a Haar subspace; (b) either K is an alternation singleton or H is a Haar subspace and K has property \(B_ 1\); (c) for K not a singleton, H is a Haar subspace. For example, the property in (c) is equivalent to property \(\tilde C\).