Characterization of generalized convex functions by best L²-approximations

Let M be a normed linear space, and {M_n}₁^∞ a sequence of increasing finite dimensional subspaces, i.e., M_n ⊂ M_{n + 1}, for all n. For any element f{hook} ε{lunate} M, we obviously have d(f{hook}, M_n) ≥ d(f{hook}, M_{n + 1}), for all n, (1) where d(f{hook}, M_k) is the distance, in the metric induced by the norm, from M_k to f{hook}. In a recent paper [2], we discussed the space M = C[a,b] with the uniform norm and with M_n = [u₀,..., u_{n - 1}], the linear subspace spanned by {u_i}₀^{n - 1}, where {u_i}₀^∞ is an infinite Tchebycheff system. We established there that the functions for which inequality (1), for a given n, is strict for all subintervals of [a, b] are precisely those that are convex with respect to (u₀, u₁,..., u_{n - 1}). The proof depended crucially on the alternance properties of the best approximants in the uniform norm. Somewhat surprisingly, analogous results are valid when the norm under consideration is the L²-norm. In fact, as we show in this paper, generalized convex functions play the same role in the L²-norm, for all continuous weights. Weaker results for the L² case were obtained in [6].