Coconvex Approximation in the Uniform Norm: the Final Frontier

Our main interest in this paper is approximation of a continuous function, on a finite interval, which changes convexity finitely many times by algebraic polynomials which are coconvex with it. This topic has received much attention in recent years, and the purpose of this paper is to give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik (D-T) moduli of smoothness. Let C[a, b] denote the space of continuous functions f on [a, b], equipped with the uniform norm ‖f‖[a,b] := maxx∈[a,b] |f(x)|. When dealing with the generic interval [−1, 1], we omit the special reference to the interval, namely, we write ‖f‖ := ‖f‖[−1,1]. To make the notion of coconvexity more precise we first denote by Ys, s ≥ 1, the set of all collections Ys := {yi}i=1, such that ys+1 := −1 < ys < . . . < y1 < 1 =: y0, and Y0 := {∅}. Let ∆(Ys) denote the collection of all functions f ∈ C[−1, 1] that change convexity at the points of the set Ys, and are convex in [y1, 1]. In particular, ∆ 2 := ∆(Y0) is the set of all convex functions f ∈ C[−1, 1]. Also with Π(x) := ∏si=1(x − yi), if f ∈ C2(−1, 1) ∩ C[−1, 1], then f ∈ ∆(Ys) if and only if f (x)Π(x) ≥ 0, x ∈ (−1, 1). (1.1)