Negative result in pointwise 3-convex polynomial approximation - Bondarenko, A. V.; Gilewicz, J. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
Let Δ^3 be the set of functions three times continuously differentiable on [−1, 1] and such that f’’’(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1] such that ∥f (r)∥_C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ^3 [−1, 1], there exists x such that |f(x)−P(x)| ≥ C nρ^r_n(x), where C > 0 is a constant that depends only on r, ρ_n(x) := 1n^2 + 1n 1−x^2.