Negative result in pointwise 3-convex polynomial approximation
Abstract
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 \sqrt{n}ρ^r_n(x), where C > 0 is a constant that depends only on r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}.Downloads
Published
25.04.2009
Issue
Section
Short communications
How to Cite
Bondarenko, A. V., and J. Gilewicz. “Negative Result in Pointwise 3-Convex Polynomial Approximation”. Ukrains’kyi Matematychnyi Zhurnal, vol. 61, no. 4, Apr. 2009, pp. 563-7, https://umj.imath.kiev.ua/index.php/umj/article/view/3040.