Functions with strictly decreasing distances from increasing Tchebycheff subspaces

AbstractLet {ui}i = 0∞ be a sequence of continuous functions on [0, 1] such that (u0,…, uk) is a Tchebycheff system on [0, 1] for all k ⩾ 0 and let C(u0,…, uk) denote the corresponding generalized convexity cone. It is proved that if f belongs to C(u0,…, un − 1), then its distance from the linear space spanned by (u0,…, un) is strictly smaller than its distance from the linear space spanned by (u0,…, un − 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 C(u0,…, un − 1) for all of these