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Izv. Akad. Nauk SSSR Ser. Mat., 1981, Volume 45, Issue 1, Pages 3–22 (Mi izv1545)  

This article is cited in 38 scientific papers (total in 38 papers)

On integral inequalities for trigonometric polynomials and their derivatives

V. V. Arestov


Abstract: Let $\Phi^+$ be the set of nondecreasing functions $\varphi$ defined on $(0,\infty)$ which admit a representation $\varphi(u)=\psi(\ln u)$, where the function $\psi$ is convex (below) on $(-\infty,\infty)$. To the class $\Phi^+$ belong, for example, the functions $\ln u$, $\ln^+u$, $u^p$ when $p>0$, and also any function $\varphi$ which is convex on $(0,\infty)$. In this paper it is shown, in particular, that if $\varphi\in\Phi^+$, then for any trigonometric polynomial $T_n$ of order $n$ the following inequality holds for all natural numbers $r$:
$$ \int_0^{2\pi}\varphi(|T_n^{(r)}(t)|) dt\leqslant\int_0^{2\pi}\varphi(n^r|T_n(t)|) dt. $$
This inequality may be considered a generalization of the inequalities of S. N. Bernstein and A. Zygmund.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1982, 18:1, 1–17

Bibliographic databases:

UDC: 517.518
MSC: 42A05
Received: 24.09.1978

Citation: V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives”, Izv. Akad. Nauk SSSR Ser. Mat., 45:1 (1981), 3–22; Math. USSR-Izv., 18:1 (1982), 1–17

Citation in format AMSBIB
\Bibitem{Are81}
\by V.~V.~Arestov
\paper On~integral inequalities for trigonometric polynomials and their derivatives
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1981
\vol 45
\issue 1
\pages 3--22
\mathnet{http://mi.mathnet.ru/izv1545}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=607574}
\zmath{https://zbmath.org/?q=an:0538.42001|0517.42001}
\transl
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 1
\pages 1--17
\crossref{https://doi.org/10.1070/IM1982v018n01ABEH001375}


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    This publication is cited in the following articles:
    1. Q.I Rahman, G Schmeisser, “Lp inequalities for polynomials”, Journal of Approximation Theory, 53:1 (1988), 26  crossref
    2. Z Ditzian, “Multivariate Bernstein and Markov inequalities”, Journal of Approximation Theory, 70:3 (1992), 273  crossref
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    5. R. M. Trigub, “Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series”, Sb. Math., 188:4 (1997), 621–638  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. I. Kozko, “Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms”, Izv. Math., 62:6 (1998), 1189–1206  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. Robert Gardner, Amy Weems, “A Bernstein TypeLpInequality for a Certain Class of Polynomials”, Journal of Mathematical Analysis and Applications, 219:2 (1998), 472  crossref
    8. P. Yu. Glazyrina, “The Markov Brothers Inequality in $L_0$-Space on an Interval”, Math. Notes, 78:1 (2005), 53–58  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. P. Yu. Glazyrina, “Markov–Nikol'skii inequality for the spaces $L_q$, $L_0$ on a segment”, Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 2, S104–S116  mathnet  mathscinet  zmath  elib
    10. È. A. Storozhenko, “Nikol'skii-Stechkin inequality for trigonometric polynomials in $L_0$”, Math. Notes, 80:3 (2006), 403–409  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. A. Aziz, W. M. Shah, “Some integral mean estimates for polynomials”, Analys in Theo Applic, 23:2 (2007), 101  crossref  mathscinet
    12. A. B. Aleksandrov, “Spectral subspaces of $L^p$ for $p<1$”, St. Petersburg Math. J., 19:3 (2008), 327–374  mathnet  crossref  mathscinet  zmath  isi
    13. R. R. Akopian, “Optimal recovery of functions analytical in a half-plane”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S1–S11  mathnet  crossref  elib
    14. M.A. Qazi, “An inequality for polynomials”, Journal of Mathematical Analysis and Applications, 336:2 (2007), 1456  crossref
    15. St. Petersburg Math. J., 21:3 (2010), 365–405  mathnet  crossref  mathscinet  zmath  isi
    16. Rajesh Pereira, “Product inequalities for norms of linear factors”, Journal of Mathematical Analysis and Applications, 356:1 (2009), 208  crossref
    17. M.A. Qazi, Q.I. Rahman, “A question concerning a polynomial inequality and an answer”, Nonlinear Analysis: Theory, Methods & Applications, 71:12 (2009), e2710  crossref
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    19. A. V. Parfenenkov, “Tochnoe neravenstvo mezhdu ravnomernymi normami algebraicheskogo mnogochlena i ego veschestvennoi chasti na kontsentricheskikh okruzhnostyakh kompleksnoi ploskosti”, Tr. IMM UrO RAN, 16, no. 4, 2010, 254–263  mathnet  elib
    20. M. A. Qazi, Q. I. Rahman, “An L 2 inequality for rational functions”, Complex Variables & Elliptic Equations, 55:7 (2010), 657  crossref
    21. Abdul Aziz, N.A.. Rather, “L<sup>p</sup> Inequalities for Polynomials”, AM, 02:03 (2011), 321  crossref
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    24. Arestov V.V., Glazyrina P.Yu., “Integralnye neravenstva dlya algebraicheskikh i trigonometricheskikh polinomov”, Doklady Akademii nauk, 442:6 (2012), 727–727  elib
    25. A. O. Leonteva, “Neravenstvo Bernshteina v $L_0$ dlya proizvodnoi nulevogo poryadka trigonometricheskikh polinomov”, Tr. IMM UrO RAN, 19, no. 2, 2013, 216–223  mathnet  mathscinet  elib
    26. 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
    27. È. A. Storozhenko, L. G. Kovalenko, “Inequality for Fractional Integrals of Complex Polynomials in $L_0$”, Math. Notes, 96:4 (2014), 609–612  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. I.E.. Simonov, P.Y.u. Glazyrina, “Sharp Markov–Nikol’skii inequality with respect to the uniform norm and the integral norm with Chebyshev weight”, Journal of Approximation Theory, 2014  crossref
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    33. A. O. Leont'eva, “Bernstein's Inequality for the Weyl Derivatives of Trigonometric Polynomials in the Space $L_0$”, Math. Notes, 104:2 (2018), 263–270  mathnet  crossref  crossref  mathscinet  isi  elib
    34. A. O. Leonteva, “Neravenstvo Bernshteina - Sege dlya proizvodnoi Veilya trigonometricheskikh polinomov v prostranstve $L_0$”, Tr. IMM UrO RAN, 24, no. 4, 2018, 199–207  mathnet  crossref  elib
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  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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