On strong unicity of best approximation in C(R)

Abstract

In the present note we give a necessary and sufficient condition on a Haar subspace which guarantees the strong unicity of best approximation in the space C(R) endowed with a certain metric Let C(R) denote the space of real valued continuous functions on the real line R endowed with the metric where h e C(R) is a given positive even function such that Furthermore, let Un be an n-dimensional Haar subspace of C(R), that is each p eUn not identically zero has at most n – 1 zeros on R. The problem of best approximation in metric of this type was investigated by G.Albinus [1]. We say that p0 e Un is a best approximation of f e C(R) if d(f,p0)≤ d(f,p) for each p e Un The following alternation theorem gives a characterization of best approximation.