The Polya algorithm for convex approximation

AbstractLet X = C[0, 1] and let b be the set of continuous convex functions on [0, 1]. If ƒ ϵ X, then the set μ∞(ƒ∣C) of all best L∞-approximants to ƒ from b is not empty and may contain more than one element. In this paper we define an element ƒ∗ in μ∞(ƒ∣C called the strict approximation to ƒ from b, and we show that limp → ∞ ƒp(x) = ƒ∗(x) for all x in [0, 1], where ƒp is the unique best Lp-approximant to ƒ from b. We then establish the continuity of the mapping ƒ → ƒ∗. Finally, we show that if ƒ, g ϵ X are uniformly close on [0, 1], and if I is a closed subinterval of (0, 1), then ƒp and gp are uniformly close on I for p ⩾ 1