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Matematicheskie Zametki, 1975, Volume 18, Issue 4, Pages 489–498 (Mi mzm9963)  

This article is cited in 19 scientific papers (total in 19 papers)

Certain inequalities in various metrics for trigonometric polynomials and their derivatives

V. I. Ivanov

V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR
Abstract: We establish for 0<p<1 the analog of the Bernstein–Zygmund inequality for the derivative of a trigonometric polynomial
ππ|tn(x)|pdxcpnpππ|tn(x)|pdx.
We prove weighted inequalities, exact in the sense of order, for trigonometric polynomials and their derivatives in various integral metrics with exponents 0<p, q.
Received: 20.12.1974
English version:
Mathematical Notes, 1975, Volume 18, Issue 4, Pages 880–885
DOI: https://doi.org/10.1007/BF01153038
Bibliographic databases:
Document Type: Article
UDC: 517.51
Language: Russian
Citation: V. I. Ivanov, “Certain inequalities in various metrics for trigonometric polynomials and their derivatives”, Mat. Zametki, 18:4 (1975), 489–498; Math. Notes, 18:4 (1975), 880–885
Citation in format AMSBIB
\Bibitem{Iva75}
\by V.~I.~Ivanov
\paper Certain inequalities in various metrics for trigonometric polynomials and their derivatives
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 4
\pages 489--498
\mathnet{http://mi.mathnet.ru/mzm9963}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=412707}
\zmath{https://zbmath.org/?q=an:0317.42001}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 4
\pages 880--885
\crossref{https://doi.org/10.1007/BF01153038}
Linking options:
  • https://www.mathnet.ru/eng/mzm9963
  • https://www.mathnet.ru/eng/mzm/v18/i4/p489
  • This publication is cited in the following 19 articles:
    1. G. A. Akishev, “Neravenstvo raznykh metrik Nikolskogo dlya trigonometricheskikh polinomov v prostranstve so smeshannoi nesimmetrichnoi normoi”, Tr. IMM UrO RAN, 29, no. 4, 2023, 11–26  mathnet  crossref  elib
    2. V. V. Arestov, M. V. Deikalova, “A Generalized Translation Operator Generated by the Sinc Function on an Interval”, Proc. Steklov Inst. Math. (Suppl.), 323, suppl. 1 (2023), S32–S52  mathnet  crossref  crossref  elib
    3. D. V. Gorbachev, “Tochnye neravenstva Bernshteina — Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2021), 58–110  mathnet  crossref
    4. Gabdolla AKİSHEV, Lars Erik PERSSON, Harpal SİNGH, “Some New Fourier and Jackson-Nikol'skii Type Inequalities In Unbounded Orthonormal Systems”, Constructive Mathematical Analysis, 4:3 (2021), 291  crossref
    5. G. A. Akishev, “Neravenstvo raznykh metrik v obobschennom prostranstve Lorentsa”, Tr. IMM UrO RAN, 24, no. 4, 2018, 5–18  mathnet  crossref  elib
    6. 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  isi  elib  scopus
    7. Simonov I.E., Glazyrina P.Yu., “Sharp Markcov-Nikol'Skii Inequality With Respect To the Uniform Norm and the Integral Norm With Chebyshev Weight”, J. Approx. Theory, 192 (2015), 69–81  crossref  isi
    8. 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  isi
    9. I. E. Simonov, “A Sharp Markov Brothers-Type Inequality in the Spaces $L_\infty$ and $L_1$ on the Segment”, Math. Notes, 93:4 (2013), 607–615  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. 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
    11. M. V. Deikalova, V. V. Rogozina, “Neravenstvo Dzheksona–Nikolskogo mezhdu ravnomernoi i integralnoi normami algebraicheskikh mnogochlenov na evklidovoi sfere”, Tr. IMM UrO RAN, 18, no. 4, 2012, 162–171  mathnet  elib
    12. V. V. Arestov, “Sharp inequalities for trigonometric polynomials with respect to integral functionals”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S21–S36  mathnet  crossref  isi  elib
    13. M. V. Deikalova, “About the sharp Jackson–Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S129–S142  mathnet  crossref  isi  elib
    14. P. Yu. Glazyrina, “The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval”, Math. Notes, 84:1 (2008), 3–21  mathnet  crossref  crossref  mathscinet  isi
    15. 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
    16. 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
    17. È. A. Storozhenko, “A problem of Mahler on the zeros of a polynomial and its derivative”, Sb. Math., 187:5 (1996), 735–744  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives”, Math. USSR-Izv., 18:1 (1982), 1–17  mathnet  crossref  mathscinet  zmath
    19. V. I. Ivanov, “Direct and converse theorems of the theory of approximation in the metric of $L_p$ for $0<p<1$”, Math. Notes, 18:5 (1975), 972–982  mathnet  mathnet  crossref
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