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Inequalities for trigonometric polynomials in spaces with integral metric

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Ukrainian Mathematical Journal Aims and scope

In the spaces L ψ (T ) of periodic functions with metric

$$ \rho {\left( {f,0} \right)_\psi } = \int\limits_T {\psi \left| {f(x)} \right|dx,} $$

where ψ is a function of the modulus-of-continuity type, we investigate analogs of the classic Bern-stein inequalities for the norms of derivatives and increments of trigonometric polynomials.

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References

  1. S. A. Pichugov, “On the Jackson theorem for periodic functions in spaces with integral metric,” Ukr. Mat. Zh., 52, No. 1, 122–133 (2000); English translation: Ukr. Math. J., 52, No. 1, 133– 147 (2000).

    Article  MathSciNet  Google Scholar 

  2. S. A. Pichugov, “On the Jackson theorem for periodic functions in metric spaces with integral metric. II,” Ukr. Mat. Zh., 63, No. 11, 1524–1533 (2011); English translation: Ukr. Math. J., 63, No. 11, 1733–1744 (2012).

    Article  MathSciNet  Google Scholar 

  3. A. F. Timan, Approximation Theory of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  4. N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  5. É. A. Storozhenko, P. Oswald, and V. G. Krotov, “Direct and inverse theorems of the Jackson type in the spaces L p , 0 < p < 1,” Mat. Stud., 98, No. 3, 395–415 (1975).

    Google Scholar 

  6. V. I. Ivanov, “Some inequalities for trigonometric polynomials and their derivatives in different metrics,” Mat. Zametki, 18, No. 4, 489–498 (1975).

    MathSciNet  MATH  Google Scholar 

  7. V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 1, 3–22 (1982).

    MathSciNet  Google Scholar 

  8. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  9. E. M. Stein and G. Weiss, Introduction to Fourier Analysis of Euclidean Spaces, Princeton University, Princeton (1971).

    Google Scholar 

  10. S. M. Nikol’skii, “Inequalities for entire functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).

    Google Scholar 

  11. N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  12. V. V. Arestov, “On the inequality of different metrics for trigonometric polynomials,” Mat. Zametki, 27, No. 4, 539–547 (1980).

    MathSciNet  MATH  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 12, pp. 1657–1671, December, 2011.

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Pichugov, S.A. Inequalities for trigonometric polynomials in spaces with integral metric. Ukr Math J 63, 1883–1899 (2012). https://doi.org/10.1007/s11253-012-0619-z

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  • DOI: https://doi.org/10.1007/s11253-012-0619-z

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