Uniform strong unicity constants for subsets of C(X)
Let X be a compact metric space and C(X) the space of continuous real- valued functions on X, with the uniform norm. If M is a Haar subspace of C(X), and \(S\subseteq C(X)\) then \(\gamma\) is a uniform strong unicity constant for S relative to M if for each \(f\in S\) \(\| f-m\| \geq \| f-B(f)\| +\gamma \| B(f)-m\|\) for all \(m\in M\), where B(f) is the best approximation to f from M. The principal Theorem 8 of the paper is for the case X an interval [a,b] and states a necessary and sufficient condition for a set \(S\subseteq C([a,b])\setminus M\) not to have a uniform strong unicity constant. The condition is in terms of sets \(E^ 0\) which are limits in the Hausdorff metric of sequences \(E(f_ k)=\{x:| f_ k(x)-B(f_ k)(x)| =\| f_ k-B(f_ k)\| \}\), \(k=1,2,..\). where \((f_ k)\) is a sequence in S. If \(a>0\), \(b\geq 0\) are integers and \(n_ k=ak+b\) for \(k=1,2,..\). then \(f(t,x)=\sum t^ k\cos (n_ k \cos^{-1}x)\) is a rational function in x and t. \textit{T. J. Rivlin} [Numer. Math. 4, 345-349 (1962; Zbl 0112.353)] studied best approximation from \(M=\Pi_ n\), the set of polynomials of degree \(\leq n\), to functions in \(S=\{f(t,\cdot):0<t<1\}\subseteq C([-1,1])\). The general results of this paper are used in showing that S does not have a uniform strong unicity constant relative to \(\Pi_ n\).
scientific article; zbMATH DE number 4111268
scientific article