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Matematicheskie Zametki, 2009, Volume 85, Issue 5, Pages 737–744
DOI: https://doi.org/10.4213/mzm4299
(Mi mzm4299)
 

This article is cited in 3 scientific papers (total in 3 papers)

Exact Constants in Generalized Inequalities for Intermediate Derivatives

A. A. Lunev, L. L. Oridoroga

Donetsk National University
Full-text PDF (457 kB) Citations (3)
References:
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:
$$ \|f\|_1^2=\|f\|_{L_2}^2+\|f^{(n)}\|_{L_2}^2\qquad\text{and}\qquad \|f\|_2^2=\sum_{l=0}^n\|f^{(l)}\|_{L_2}^2. $$
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.
Received: 19.11.2007
Revised: 02.12.2008
English version:
Mathematical Notes, 2009, Volume 85, Issue 5, Pages 703–711
DOI: https://doi.org/10.1134/S0001434609050101
Bibliographic databases:
UDC: 517.518.26
Language: Russian
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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\paper Exact Constants in Generalized Inequalities for Intermediate Derivatives
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\pages 737--744
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  • This publication is cited in the following 3 articles:
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