Advertisement

Mathematical Notes

, Volume 106, Issue 3–4, pp 412–422 | Cite as

Generalized Smoothness and Approximation of Periodic Functions in the Spaces Lp, 1 < p < +∞

  • K. V. RunovskiiEmail author
Article
  • 9 Downloads

Abstract

Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces Lp, 1 < p < +∞. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator ψ, sufficient constructive ψ-smoothness conditions, estimates of best approximations of ψ-derivatives, estimates of best approximations of ψ-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.

Keywords

best approximation modulus of smoothness generalized derivative 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The author wishes to express gratitude to the referee for valuable and useful remarks that have led to the improvement of the paper.

References

  1. 1.
    A. I. Stepanets, Classification and Approximation of Periodic Functions (Naukova Dumka, Kiev, 1987) [in Russian].zbMATHGoogle Scholar
  2. 2.
    H. Triebel, Higher Analysis (Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992).Google Scholar
  3. 3.
    K. Runovski and H.-J. Schmeisser, “Smoothness and function spaces generated by homogeneous multipliers,” J. Fund Spaces Appl., No. Art. ID 643135 (2012).zbMATHGoogle Scholar
  4. 4.
    H.-J. Schmeisser and W. Sickel, “Characterization periodic function spaces via means of Abel-Poisson and Bessel-potential type,” J. Approx. Theory 61 (2), 239–262 (1990).MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. V. Runovskii, “A direct theorem of approximation theory for a general modulus of smoothness,” Mat. Zametki 95 (6), 899–910 (2014) [Math. Notes 95 (6), 833–842 (2014)].MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. V. Runovskii, “Approximation by Fourier means and generalized moduli of smoothness,” Mat. Zametki 99 (4), 574–587 (2016) [Math. Notes 99 (4), 564–575 (2016)].MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. V. Runovskii, “Trigonometric polynomial approximation, K-functional and generalized moduli of smoothness,” Mat. Sb. 208(2), 70–87 (2017) [Sb. Math. 208(2), 237–254 (2017)].MathSciNetCrossRefGoogle Scholar
  8. 8.
    K. Runovski and H.-J. Schmeisser, “General moduli of smoothness and approximation by families of linear polynomial operators,” in New Perspectives on Approximation and Sampling Theory (Birkhauser, Cham, 2014), pp. 269–298.CrossRefGoogle Scholar
  9. 9.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Eucliadean Spaces (Princeton Univ. Press, Princeton, NJ, 1971).Google Scholar
  10. 10.
    P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Vol. 1. One-Dimensional Theory (Academic Press, New York, 1971).CrossRefGoogle Scholar
  11. 11.
    A. I. Stepanets, Methods of Approximation Theory. I, II, in Trudy. Inst. Mat. NAN Ukr, Vol. 40: Mathematics and Its Applications (Inst. Mat. NAN Ukr., Kiev, 2002), [in Russian].zbMATHGoogle Scholar
  12. 12.
    S. B. Stechkin, “A generalization of some inequalities of S. N. Bernstein,” Dokl. AN SSSR 60, 1511–1514 (1948).MathSciNetzbMATHGoogle Scholar
  13. 13.
    V. I. Ivanov, “Direct and converse theorems of the theory of approximation in the metric of L p for 0 < p < 1,” Mat. Zametki 18 (5), 641–658 (1975) [Math. Notes 18(5), 972–982(1975)].MathSciNetGoogle Scholar
  14. 14.
    E. A. Storozhenko, V. G. Kjotov, and P. Oswald, “Direct and converse theorems of Jackson type and the inverse theorems of Jackson type in L p spaces, 0 p < 1,” Mat. Sb. 98 (140) (3(11)), 395–415 (1975) [Math. USSR-Sb. 27(3)355–374(1975)].MathSciNetGoogle Scholar
  15. 15.
    V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives,” Izv Akad. Nauk SSSR Ser.Mat. 45(1), 3–22 (1981) [Math. USSR-Izv. 18(1), 1–17(1982)].MathSciNetGoogle Scholar
  16. 16.
    V M. Tikhomirov, Some Questions in Approximation Theory (Izd. Moskov Univ., Moscow, 1976) [in Russian].Google Scholar
  17. 17.
    E. Belinski and E. Liflyand, “Approximation properties in L p, 0 < p < 1,” Funct. Approx. Comment. Math. 22, 189–199 (1994).Google Scholar
  18. 18.
    K. Runovski and H.-J. Schmeisser, “On some extensions of Bernstein's inequalities for trigonometric polynomials,” Funct. Approx. Comment. Math. 29, 125–142 (2001).MathSciNetCrossRefGoogle Scholar
  19. 19.
    B. V. Simonov and S. Yu. Tikhonov, “Embedding theorems in constructive approximation,” Mat. Sb. 199(9), 107–148 (2008) [Sb. Math. 199(9), 1367–1407(2008)].MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. L. Butzer, H. Dyckhoff, E. Gorlich, and R. L. Stens, “Best trigonometric approximation, fractional order derivatives and Lipschitz classes,” Canadian J. Math. 29 (4), 781–793 (1977).MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. K. Potapov and B. V. Simonov, “Moduli of smoothness of positive order of functions from the spaces L p, 1 < p < +∞,” in Trudy Mekh.-Mat. Fak. MGU, Modern Problems of Mathematics and Mechanics (Izd. Mekh.-Mat. Fak. MGU, Moscow, 2011), Vol. 7, No. 1 pp. 100–109, [in Russian].Google Scholar
  22. 22.
    K. V. Runovski and H.-J. Schmeisser, “Moduli of smoothness related to fractional Riesz-derivatives,” Z. Anal. Anwend. 34(1), 109–125 (2015).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations