Skip to main content
SpringerLink
Log in

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

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MathSciNet  Google Scholar 

  2. Bergh, J., Löfström, J.: Interpolation Spaces. Springer-Verlag (Berlin-Heidelberg, An Introduction (1976)

    Book  Google Scholar 

  3. Bergh, J., Peetre, J.: On the spaces \(V_p\) (\(0<p<\infty \)). Boll. Unione Mat. Ital. Ser. 4(10), 632–648 (1974)

    MATH  Google Scholar 

  4. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Vol. 1, Academic Press (New York–London, 1971)

  5. Gogoladze, L., Meskhia, R.: On the absolute convergence of trigonometric Fourier series. Proc. Razmadze Math. Inst. 141, 29–40 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Gorbachev, D., Liflyand, E., Tikhonov, S.: Weighted Fourier inequalities: Boas conjecture in \(\mathbb{R}^n\). J. Anal. Math. 114, 99–120 (2011)

    Article  MathSciNet  Google Scholar 

  7. Gorbachev, D., Tikhonov, S.: Moduli of smoothness and growth prperties of Fourier transforms: two-sided estimates. J. Approx. Theory 164, 1283–1312 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Univ, Press (Cambridge (1934)

    MATH  Google Scholar 

  9. E. Liflyand and S. Tikhonov, Extended solution of Boas' conjecture on Fourier transforms, C.R. Acad. Sci. Paris, Ser. I., 346 (2008), 1137–1142

    Article  MathSciNet  Google Scholar 

  10. Móricz, F.: Sufficient conditions for the Lebesgue integrability of Fourier transforms. Anal. Math. 36, 121–129 (2010)

    Article  MathSciNet  Google Scholar 

  11. S. M. Nikols'kii, Approximation of Functions of Several Variables and Embedding Theorems, Springer (Berlin–Heidelberg, 1975)

  12. Onneweer, C.W.: On absolutely convergent Fourier series. Arkiv Mat. 12, 51–58 (1974)

    Article  MathSciNet  Google Scholar 

  13. J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Depart. (Durham, USA, 1976)

  14. Sagher, Y.: Integrability conditions for the Fourier transform. J. Math. Anal. Appl. 54, 151–156 (1976)

    Article  MathSciNet  Google Scholar 

  15. A. P. Terekhin, Approximation of functions of bounded \(p\)-variation, Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], 2 (1965), 171–187

  16. Terekhin, A.P.: Integral smoothness properties of periodic functions of bounded \(p\)-variation. Math. Notes 2, 659–665 (1967)

    Article  MathSciNet  Google Scholar 

  17. A. F. Timan, Theory of Approximation of Functions of a Real Variable, Macmillan (New York, 1963)

  18. E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press (Oxford, 1937)

  19. W. Trebels, Estimates for Moduli of Continuity of Functions Given by Their Fourier Transform, Lecture Notes in Math., vol. 571, Springer (Berlin, 1977)

    Chapter  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  21. Volosivets, S.S.: Convergence of series of Fourier coefficients of \(p\)-absolutely continuous functions. Anal. Math. 26, 63–80 (2000)

    Article  MathSciNet  Google Scholar 

  22. Wiener, N.: The quadratic variation of a function and its Fourier coefficient. J. Math. Phys. 3, 72–94 (1924)

    Article  Google Scholar 

  23. Young, L.C.: An inequality of Hölder type connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)

    Article  MathSciNet  Google Scholar 

  24. A. Zygmund, Trigonometric Series, Vol. I., Cambridge Univ. Press (1959)

Download references

Acknowledgement

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

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Saratov State University, Astrakhanskaya Str. 83, 410012, Saratov, Russia

    S. A. Krayukhin & S. S. Volosivets

Authors
  1. S. A. Krayukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. S. Volosivets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. S. Volosivets.

Additional information

Dedicated to our teacher, Professor B. I. Golubov on his eightieth birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Krayukhin, S.A., Volosivets, S.S. Functions of bounded p-variation and weighted integrability of Fourier transforms. Acta Math. Hungar. 159, 374–399 (2019). https://doi.org/10.1007/s10474-019-00995-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-019-00995-6

Key words and phrases

Mathematics Subject Classification

Search

Navigation