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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 3, Pages 29–38 (Mi faa115)  

This article is cited in 10 scientific papers (total in 10 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
$$ |f^{(k)}(0)|\le A_{n,k}(\int_0^{+\infty}(|f(x)|^2+|f^{(n)}(x)|^2) dx)^{1/2}. $$
Specifically, we prove that
$$ A_{n,k}=(\sin\frac{\pi(2k+1)}{2n})^{-1/2} \prod_{s=1}^k\operatorname{cot}\frac{\pi s}{2n}  $$
for all $n\in\{1,2,…\}$ and $k\in\{0,…,n-1\}$. 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

DOI: https://doi.org/10.4213/faa115

Full text: PDF file (197 kB)
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English version:
Functional Analysis and Its Applications, 2004, 38:3, 184–191

Bibliographic databases:

UDC: 517.518.26
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

Citation in format AMSBIB
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\by G.~A.~Kalyabin
\paper Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)
\jour Funktsional. Anal. i Prilozhen.
\yr 2004
\vol 38
\issue 3
\pages 29--38
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\crossref{https://doi.org/10.4213/faa115}
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\zmath{https://zbmath.org/?q=an:1079.41009}
\transl
\jour Funct. Anal. Appl.
\yr 2004
\vol 38
\issue 3
\pages 184--191
\crossref{https://doi.org/10.1023/B:FAIA.0000042803.72039.20}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-4644367883}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. G. A. Kalyabin, “Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis”, Proc. Steklov Inst. Math., 243 (2003), 220–226  mathnet  mathscinet  zmath
    2. G. A. Kalyabin, “Effective Formulas for Constants in the Stechkin–Gabushin Problem”, Proc. Steklov Inst. Math., 248 (2005), 118–124  mathnet  mathscinet  zmath
    3. G. A. Kalyabin, “Some Problems for Sobolev Spaces on the Half-line”, Proc. Steklov Inst. Math., 255 (2006), 150–158  mathnet  crossref  mathscinet  elib
    4. Watanabe K., Kametaka Y., Nagai A., Takemura K., Yamagishi H., “The best constant of Sobolev inequality on a bounded interval”, J. Math. Anal. Appl., 340:1 (2008), 699–706  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. A. Lunev, L. L. Oridoroga, “Exact Constants in Generalized Inequalities for Intermediate Derivatives”, Math. Notes, 85:5 (2009), 703–711  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Watanabe K., Kametaka Y., Nagai A., Yamagishi H., TakemuraK., “Symmetrization of functions and the best constant of 1-DIM $L^p$ Sobolev inequality”, J. Inequal. Appl., 2009, 874631, 12 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. Tikhomirov, A. Kochurov, “Kolmogorov-type inequalities on the whole line or half line and the Lagrange principle in the theory of extremum problems”, Eurasian Math. J., 2:3 (2011), 125–142  mathnet  mathscinet  zmath
    8. Oshime Y., Watanabe K., “The Best Constant of l-P Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II”, Tokyo J. Math., 34:1 (2011), 115–133  crossref  mathscinet  zmath  isi  scopus
    9. S. V. Zelik, A. A. Ilyin, “Green's function asymptotics and sharp interpolation inequalities”, Russian Math. Surveys, 69:2 (2014), 209–260  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Osipenko K.Yu., “Recovery of Derivatives For Functions Defined on the Semiaxis”, J. Complex., 48 (2018), 111–118  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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