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 Trudy Mat. Inst. Steklova, 2005, Volume 248, Pages 124–129 (Mi tm125)

Effective Formulas for Constants in the Stechkin–Gabushin Problem

G. A. Kalyabin

Image Processing Systems Institute

Abstract: Explicit and transparent expressions are found for the numbers $S_{n,k}$ involved in the formula $E(N,n,k)= S_{n,k} N^{-\beta /\alpha }$, where $\alpha :=(2k+1)/2n$, $\beta := 1-\alpha$, and $k\in \{0,1,…,n-1\}$, for the best approximation of the operators $d^k/dx^k$ in the $C(\mathbb R_+)$ metric on the class of functions $f$ such that $\|f\|_{L_2(\mathbb R_+)} <\infty$ and $\|f^{(n)}\|_{L_2(\mathbb R_+)}\le 1$ by means of linear operators $V$ whose norms satisfy the inequality $\|V\|_{L_2(\mathbb R_+)\to C(\mathbb R_+)}\le N$. Simultaneously, the values of the sharp constants $K_{n,k}$ in the Kolmogorov inequality $\|f^{(k)}\|_{C(\mathbb R_+)}\le K_{n,k}\|f^{(n)}\|^{\alpha }_{L_2(\mathbb R_+)} \|f\|^{\beta }_{L_2 (\mathbb R_+)}$ are determined. The symmetry and regularity properties of the constants, as well as their asymptotic behavior as $n\to \infty$, are studied.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 248, 118–124

Bibliographic databases:
UDC: 517.51

Citation: G. A. Kalyabin, “Effective Formulas for Constants in the Stechkin–Gabushin Problem”, Studies on function theory and differential equations, Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 248, Nauka, MAIK «Nauka/Inteperiodika», M., 2005, 124–129; Proc. Steklov Inst. Math., 248 (2005), 118–124

Citation in format AMSBIB
\Bibitem{Kal05} \by G.~A.~Kalyabin \paper Effective Formulas for Constants in the Stechkin--Gabushin Problem \inbook Studies on function theory and differential equations \bookinfo Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii \serial Trudy Mat. Inst. Steklova \yr 2005 \vol 248 \pages 124--129 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm125} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165922} \zmath{https://zbmath.org/?q=an:1122.41016} \transl \jour Proc. Steklov Inst. Math. \yr 2005 \vol 248 \pages 118--124 

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This publication is cited in the following articles:
1. G. A. Kalyabin, “Some Problems for Sobolev Spaces on the Half-line”, Proc. Steklov Inst. Math., 255 (2006), 150–158
2. Watanabe, K, “The best constant of Sobolev inequality on a bounded interval”, Journal of Mathematical Analysis and Applications, 340:1 (2008), 699
3. S. K. Bagdasarov, “Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm”, Izv. Math., 74:2 (2010), 219–279
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