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Mat. Zametki, 2008, Volume 84, Issue 1, Pages 3–22 (Mi mz5194)  

This article is cited in 7 scientific papers (total in 7 papers)

The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval

P. Yu. Glazyrina

Ural State University

Abstract: In this paper, an inequality between the $L_q$-mean of the $k$th derivative of an algebraic polynomial of degree $n\ge 1$ and the $L_0$-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for $k=0$, $q\in[0,\infty]$ and $1\le k\le n$, $q\in\{0\}\cup[1,\infty]$. Here a new method for finding the best constant for all $0\le k\le n$, $q\in[0,\infty]$, and, in particular, for the case $1\le k\le n$, $q\in(0,1)$, which has not been studied before is proposed. We find the order of growth of the best constant with respect to $n$ as $n\to \infty$ for fixed $k$ and $q$.

Keywords: algebraic polynomial, Markov–Nikolskii inequality, the spaces $L_q$ and $L_0$, geometric mean of a polynomial, $L_q$-mean, extremal polynomial, majorization principle

DOI: https://doi.org/10.4213/mzm5194

Full text: PDF file (618 kB)
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English version:
Mathematical Notes, 2008, 84:1, 3–21

Bibliographic databases:

UDC: 517.518.862
Received: 31.07.2007

Citation: P. Yu. Glazyrina, “The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval”, Mat. Zametki, 84:1 (2008), 3–22; Math. Notes, 84:1 (2008), 3–21

Citation in format AMSBIB
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\pages 3--22
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\crossref{https://doi.org/10.4213/mzm5194}
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\jour Math. Notes
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\pages 3--21
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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. I. E. Simonov, “Tochnoe neravenstvo tipa bratev Markovykh v prostranstvakh $L_p$, $L_1$ na otrezke”, Tr. IMM UrO RAN, 17, no. 3, 2011, 282–290  mathnet  elib
    2. M. R. Gabdullin, “Otsenka srednego geometricheskogo proizvodnoi mnogochlena cherez ego ravnomernuyu normu na otrezke”, Tr. IMM UrO RAN, 18, no. 4, 2012, 153–161  mathnet  elib
    3. V. V. Arestov, M. V. Deikalova, “Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 9–23  mathnet  crossref  mathscinet  isi  elib
    4. Klurman O., “On Constrained Markov-Nikolskii Type Inequalities For K-Absolutely Monotone Polynomials”, Acta Math. Hung., 143:1 (2014), 13–22  crossref  mathscinet  zmath  isi  scopus
    5. Arestov V. Deikalova M., “Nikol'Skii Inequality Between the Uniform Norm and l-Q-Norm With Ultraspherical Weight of Algebraic Polynomials on An Interval”, Comput. Methods Funct. Theory, 15:4, SI (2015), 689–708  crossref  mathscinet  zmath  isi  elib  scopus
    6. Sroka G., “Constants in Va Markov'S Inequality in l-P Norms”, J. Approx. Theory, 194 (2015), 27–34  crossref  mathscinet  zmath  isi  elib  scopus
    7. Arestov V. Deikalova M., “Nikol'skii inequality between the uniform norm and L q -norm with Jacobi weight of algebraic polynomials on an interval”, Anal. Math., 42:2 (2016), 91–120  crossref  mathscinet  zmath  isi  elib  scopus
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