TY - JOUR
AU - A. V. Bondarenko
AU - J. Gilewicz
PY - 2009/04/25
Y2 - 2023/02/05
TI - Negative result in pointwise 3-convex polynomial approximation
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 61
IS - 4
SE - Short communications
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3040
AB - 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 $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$.
ER -