Advertisement

Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 374–399 | Cite as

Functions of bounded p-variation and weighted integrability of Fourier transforms

  • S. A. Krayukhin
  • S. S. VolosivetsEmail author
Article
  • 19 Downloads

Abstract

We study some properties of functions of bounded p-variation on \(\mathbb{R}\) and its specific fractional moduli of smoothness, including the connection between p-variational and Lp best approximations and moduli of smoothness. These properties are used to derive the results concerning weighted integrability of Fourier transforms.

Key words and phrases

functions of bounded p-variation approximation by entire functions of exponential type p-variational and Lp moduli of smoothness Fourier transform weighted integrability 

Mathematics Subject Classification

41A17 41A30 42A38 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgement

The authors thank the referee for useful suggestions and remarks, which improved the revised version of our paper.

References

  1. 1.
    Bary, N.K., Stechkin, S.B.: Best approximation and differential properties of two conjugate functions. Trudy Mosk. Mat. Obs. 5, 483–522 (1956). (in Russian)MathSciNetGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation Spaces. Springer-Verlag (Berlin-Heidelberg, An Introduction (1976)CrossRefGoogle Scholar
  3. 3.
    Bergh, J., Peetre, J.: On the spaces \(V_p\) (\(0<p<\infty \)). Boll. Unione Mat. Ital. Ser. 4(10), 632–648 (1974)zbMATHGoogle Scholar
  4. 4.
    P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Vol. 1, Academic Press (New York–London, 1971)Google Scholar
  5. 5.
    Gogoladze, L., Meskhia, R.: On the absolute convergence of trigonometric Fourier series. Proc. Razmadze Math. Inst. 141, 29–40 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gorbachev, D., Liflyand, E., Tikhonov, S.: Weighted Fourier inequalities: Boas conjecture in \(\mathbb{R}^n\). J. Anal. Math. 114, 99–120 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gorbachev, D., Tikhonov, S.: Moduli of smoothness and growth prperties of Fourier transforms: two-sided estimates. J. Approx. Theory 164, 1283–1312 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Univ, Press (Cambridge (1934)zbMATHGoogle Scholar
  9. 9.
    E. Liflyand and S. Tikhonov, Extended solution of Boas' conjecture on Fourier transforms, C.R. Acad. Sci. Paris, Ser. I., 346 (2008), 1137–1142MathSciNetCrossRefGoogle Scholar
  10. 10.
    Móricz, F.: Sufficient conditions for the Lebesgue integrability of Fourier transforms. Anal. Math. 36, 121–129 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. M. Nikols'kii, Approximation of Functions of Several Variables and Embedding Theorems, Springer (Berlin–Heidelberg, 1975)Google Scholar
  12. 12.
    Onneweer, C.W.: On absolutely convergent Fourier series. Arkiv Mat. 12, 51–58 (1974)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Depart. (Durham, USA, 1976)Google Scholar
  14. 14.
    Sagher, Y.: Integrability conditions for the Fourier transform. J. Math. Anal. Appl. 54, 151–156 (1976)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. P. Terekhin, Approximation of functions of bounded \(p\)-variation, Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], 2 (1965), 171–187Google Scholar
  16. 16.
    Terekhin, A.P.: Integral smoothness properties of periodic functions of bounded \(p\)-variation. Math. Notes 2, 659–665 (1967)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. F. Timan, Theory of Approximation of Functions of a Real Variable, Macmillan (New York, 1963)Google Scholar
  18. 18.
    E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press (Oxford, 1937)Google Scholar
  19. 19.
    W. Trebels, Estimates for Moduli of Continuity of Functions Given by Their Fourier Transform, Lecture Notes in Math., vol. 571, Springer (Berlin, 1977)CrossRefGoogle Scholar
  20. 20.
    Ul'yanov, P.L.: Series with respect to a Haar system with monotone coefficients. Izv. Akad. Nauk SSSR. Ser. Mat. 28, 925–950 (1964). (in Russian)MathSciNetGoogle Scholar
  21. 21.
    Volosivets, S.S.: Convergence of series of Fourier coefficients of \(p\)-absolutely continuous functions. Anal. Math. 26, 63–80 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wiener, N.: The quadratic variation of a function and its Fourier coefficient. J. Math. Phys. 3, 72–94 (1924)CrossRefGoogle Scholar
  23. 23.
    Young, L.C.: An inequality of Hölder type connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Zygmund, Trigonometric Series, Vol. I., Cambridge Univ. Press (1959)Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsSaratov State UniversitySaratovRussia

Personalised recommendations