Negative result in pointwise 3-convex polynomial approximation
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, fo...
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| Дата: | 2009 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3040 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | 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
$r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$. |
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