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This article is cited in 13 scientific papers (total in 13 papers)
Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)
G. A. Kalyabinab a Image Processing Systems Institute
b Samara Academy of Humanities
Abstract:
We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities
$$
|f^{(k)}(0)|\le A_{n,k}\bigg(\int_0^{+\infty}(|f(x)|^2+|f^{(n)}(x)|^2)\,dx\bigg)^{1/2}.
$$
Specifically, we prove that
$$
A_{n,k}=\bigg(\sin\frac{\pi(2k+1)}{2n}\bigg)^{-1/2} \prod_{s=1}^k\operatorname{cot}\frac{\pi s}{2n}\,
$$
for all $n\in\{1,2,\dots\}$ and $k\in\{0,\dots,n-1\}$. We establish symmetry and regularity properties of the numbers $A_{n,k}$ and study their asymptotic behavior as $n\to\infty$ for the cases $k=O(n^{2/3})$ and $k/n\to\alpha\in(0,1)$.
Similar problems were previously studied by Gabushin and Taikov.
Keywords:
extrapolation with minimal norm, Lagrange optimality principle, inversion of special matrices.
Received: 16.06.2003
Citation:
G. A. Kalyabin, “Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)”, Funktsional. Anal. i Prilozhen., 38:3 (2004), 29–38; Funct. Anal. Appl., 38:3 (2004), 184–191
Linking options:
https://www.mathnet.ru/eng/faa115https://doi.org/10.4213/faa115 https://www.mathnet.ru/eng/faa/v38/i3/p29
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