2017
Том 69
№ 9

All Issues

Negative result in pointwise 3-convex polynomial approximation

Bondarenko A. V., Gilewicz J.

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

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 4, pp 674-681.

Citation Example: Bondarenko A. V., Gilewicz J. Negative result in pointwise 3-convex polynomial approximation // Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 563-567.

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