ABSTRACT. Let T be a locally compact subset of R and C_{0}(T) the space of continuous functions which vanish at infinity. An n-dimensional subspace G of C_{0}(T) may possess one of the three alternation properties:

(A-1) For each f ∈ C_{0}(T) which has a unique best approximation g_{0} ∈ G to f, f − g_{0} has n + 1 alternating peak points;

(A-2) For each f ∈ C_{0}(T), there exists a best approximation g_{0} ∈ G to f such that f − g_{0} has n + 1 alternating peak points;

(A-3) For each f ∈ C_{0}(T) and each best approximation g_{0} ∈ G to f, f − g_{0} has n + 1 alternating peak points.

In this paper, for each i ∈ {1, 2, 3} we give an intrinsic characterization of those subspaces G of C_{0}(t) which have property (A-i).