Abstract:
We establish for 0<p<1 the analog of the Bernstein–Zygmund inequality for the derivative
of a trigonometric polynomial
∫π−π|t′n(x)|pdx⩽cpnp∫π−π|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⩽∞.
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
\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:
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
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
D. V. Gorbachev, “Tochnye neravenstva Bernshteina — Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2021), 58–110
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
G. A. Akishev, “Neravenstvo raznykh metrik v obobschennom prostranstve Lorentsa”, Tr. IMM UrO RAN, 24, no. 4, 2018, 5–18
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
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
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
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
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
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
V. V. Arestov, “Sharp inequalities for trigonometric polynomials with respect to integral functionals”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S21–S36
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
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
P. Yu. Glazyrina, “The Markov Brothers Inequality in $L_0$-Space on an Interval”, Math. Notes, 78:1 (2005), 53–58
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
È. A. Storozhenko, “A problem of Mahler on the zeros of a polynomial and its derivative”, Sb. Math., 187:5 (1996), 735–744
V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives”, Math. USSR-Izv., 18:1 (1982), 1–17
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