Functions with strictly decreasing distances from increasing Tchebycheff subspaces

Let {u_i}_{i = 0}^∞ be a sequence of continuous functions on [0, 1] such that (u₀,..., u_k) is a Tchebycheff system on [0, 1] for all k ≥ 0 and let C(u₀,..., u_k) denote the corresponding generalized convexity cone. It is proved that if f belongs to C(u₀,..., u_{n - 1}), then its distance from the linear space spanned by (u₀,..., u_n) is strictly smaller than its distance from the linear space spanned by (u₀,..., u_{n - 1}). Other properties of the best approximants to such functions are also given. It is shown, by a general category argument, that no direct converse can exit. It is then established that if strict decrease of distances (or one of a number of other properties of the best approximants) holds for all subintervals of [0, 1], then f ε{lunate} C(u₀,..., u_{n - 1}) for all of these.