Matematicheskie Zametki
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (457 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.518.26
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
\Bibitem{LunOri09} \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 \mathnet{http://mi.mathnet.ru/mz4299} \crossref{https://doi.org/10.4213/mzm4299} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2572863} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267684500010} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70049096252} 

• http://mi.mathnet.ru/eng/mz4299
• https://doi.org/10.4213/mzm4299
• http://mi.mathnet.ru/eng/mz/v85/i5/p737

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 403 Full text: 205 References: 42 First page: 14