Journal of Mathematical Sciences

, Volume 187, Issue 1, pp 49–56 | Cite as

Fourier transform versus Hilbert transform

  • Elijah Liflyand


Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.


Fourier transform integrability Hilbert transform Hardy space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Aubertin and J. J. F. Fournier, “Integrability theorems for trigonometric series,” Stud. Math., 107, 33–59 (1993).MathSciNetMATHGoogle Scholar
  2. 2.
    N. K. Bary, A Treatise on Trigonometric Series, Macmillan, New York, 1964.MATHGoogle Scholar
  3. 3.
    E. S. Belinsky, “On asymptotic behavior of integral norms of trigonometric polynomials,” in Metric Questions of the Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev, 1975, pp. 15–24.Google Scholar
  4. 4.
    S. V. Bochkarev, “On a problem of Zygmund,” Math. USSR-Izv., 7, 629-–637 (1973).CrossRefGoogle Scholar
  5. 5.
    S. Fridli, “Hardy Spaces Generated by an Integrability Condition,” J. Approx. Theory, 113, 91–109 (2001).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.MATHGoogle Scholar
  7. 7.
    R. L. Johnson and C. R. Warner, “The convolution algebra H 1(R),J. Funct. Spaces Appl., 8, 167–179 (2010).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    J.-P. Kahane, Séries de Fourier Absolument Convergentes, Springer, Berlin, 1970.MATHGoogle Scholar
  9. 9.
    H. Kober, “A note on Hilbert’s operator,” Bull. Amer. Math. Soc., 48:1, 421–426 (1942).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, Singapore, 1991.MATHCrossRefGoogle Scholar
  11. 11.
    E. Liflyand, “Fourier transforms of functions from certain classes,” Anal. Math., 19, 151–168 (1993).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    E. Liflyand and S. Tikhonov, “The Fourier transforms of general monotone functions,” in Analysis and Mathematical Physics, Birkhäuser, Basel, 2009, pp. 373–391.Google Scholar
  13. 13.
    E. Liflyand and S. Tikhonov, “Weighted Paley–Wiener theorem on the Hilbert transform,” C. R. Acad. Sci. Paris, Ser. I, 348, 1253–1258 (2010).Google Scholar
  14. 14.
    E. Liflyand and S. Tikhonov, “A concept of general monotonicity and applications,” Math. Nachr., 284, 1083–1098 (2011).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    R. E. A. C. Paley and N. Wiener, “Notes on the theory and application of Fourier transforms,” Trans. Amer. Math. Soc., 35, 348–355 (1933).MathSciNetGoogle Scholar
  16. 16.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.MATHGoogle Scholar
  17. 17.
    E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937.Google Scholar
  18. 18.
    R. M. Trigub, “Multipliers of Fourier series and approximation of functions by polynomials in spaces C and L,Soviet Math. Dokl., 39, 494–498 (1989).MathSciNetMATHGoogle Scholar
  19. 19.
    R. M. Trigub, “A Generalization of the Euler-Maclaurin formula,” Math. Notes, 61, 253–257 (1997).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    R. M. Trigub and E. S. Belinsky, Fourier Analysis and Appoximation of Functions, Kluwer, Dordrecht, 2004.Google Scholar
  21. 21.
    N. Wiener, The Fourier Integral and Certain of Its Applications, Dover, New York, 1959.MATHGoogle Scholar
  22. 22.
    A. Zygmund, “Some Points in the theory of trigonometric and power series,” Trans. Amer. Math. Soc., 36, 586–617 (1934).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

Personalised recommendations