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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
\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
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 1
\pages 1--17

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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
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    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
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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
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    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
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