Negative result in pointwise 3-convex polynomial approximation

  • A. V. Bondarenko
  • J. Gilewicz

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 $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$.
Published
25.04.2009
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.
Section
Short communications