Geodesic
Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 29-38 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {G. A. Kalyabin},
     title = {Sharp {Constants} in {Inequalities} for {Intermediate} {Derivatives} (the {Gabushin} {Case)}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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G. A. Kalyabin. Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case). Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 29-38. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a2/

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