On weak Chebyshev subspaces. I. Equioscillation of the error in approximation

This paper is a generalization of the result obtained by F. Deutsch, G. Nürnberger, and I. Singer (1980, Pacific J. Math. 88, 9-31). It is shown that if Q is a locally compact totally ordered space, and N is an n-dimensional subspace of C₀(Q), then N is a weak Chebyshev subspace if and only if for each f ε{lunate} C₀(Q), there is g ε{lunate} N such that ∥f - g∥ = d(f, N) and (f - g) equioscillates at (n + 1) points.