Coconvex Polynomial Approximation of Twice Differentiable Functions Original Research Article

For a function f ∈ C2[−1, 1] with 1 ≤ r < ∞ inflection points and sufficiently large n we construct an algebraic polynomial pn of degree ≤ n satisfying f″(x) p″n(x) ≥ 0, x ∈ [−1, 1], and such that ∥ f(ν) − p(ν)n∥∞ ≤ Cνn−2 + νωφ(f″, n− 1), ν = 0, 1, 2, where Cν = Cν(r), ν = 0, 1, C2 = C2(r)/[formula] (α is the point of inflection nearest to ±1), and ωφ(f″, n− 1) denotes the Ditzian-Totik modulus of continuity of f″ in the uniform metric.