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Mat. Zametki, 2009, Volume 85, Issue 5, Pages 737–744 (Mi mz4299)  

This article is cited in 1 scientific paper (total in 1 paper)

Exact Constants in Generalized Inequalities for Intermediate Derivatives

A. A. Lunev, L. L. Oridoroga

Donetsk National University

Abstract: Consider the Sobolev space $W_2^n(\mathbb R_+)$ on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants $A_{n,k}$ in inequalities of Kolmogorov type for the values of intermediate derivatives $|f^{(k)}(0)|\le A_{n,k}\|f\|$. In the general case, the expression for the constants $A_{n,k}$ is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants $A_{n,k}$ in the case of the following norms:
$$ \|f\|_1^2=\|f\|_{L_2}^2+\|f^{(n)}\|_{L_2}^2\qquadand\qquad \|f\|_2^2=\sum_{l=0}^n\|f^{(l)}\|_{L_2}^2. $$
In the case of the norm $\|\cdot\|_1$, formulas for the constants $A_{n,k}$ were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants $A_{n,k}$ is also studied in the case of the norm $\|\cdot\|_2$. In addition, we prove a symmetry property of the constants $A_{n,k}$ in the general case.

Keywords: Sobolev space, Kolmogorov-type inequalities, intermediate derivative, linear functional in Hilbert space, Vandermonde matrix, Cramer's rule

DOI: https://doi.org/10.4213/mzm4299

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English version:
Mathematical Notes, 2009, 85:5, 703–711

Bibliographic databases:

UDC: 517.518.26
Received: 19.11.2007
Revised: 02.12.2008

Citation: A. A. Lunev, L. L. Oridoroga, “Exact Constants in Generalized Inequalities for Intermediate Derivatives”, Mat. Zametki, 85:5 (2009), 737–744; Math. Notes, 85:5 (2009), 703–711

Citation in format AMSBIB
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\by A.~A.~Lunev, L.~L.~Oridoroga
\paper Exact Constants in Generalized Inequalities for Intermediate Derivatives
\jour Mat. Zametki
\yr 2009
\vol 85
\issue 5
\pages 737--744
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\crossref{https://doi.org/10.4213/mzm4299}
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\zmath{https://zbmath.org/?q=an:1180.41009}
\transl
\jour Math. Notes
\yr 2009
\vol 85
\issue 5
\pages 703--711
\crossref{https://doi.org/10.1134/S0001434609050101}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70049096252}


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    This publication is cited in the following articles:
    1. Osipenko K.Yu., “Recovery of Derivatives For Functions Defined on the Semiaxis”, J. Complex., 48 (2018), 111–118  crossref  mathscinet  zmath  isi  scopus
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