Advertisement

Journal of Fourier Analysis and Applications

, Volume 19, Issue 1, pp 167–179 | Cite as

On Fourier Transforms of Radial Functions and Distributions

  • Loukas Grafakos
  • Gerald Teschl
Article

Abstract

We find a formula that relates the Fourier transform of a radial function on R n with the Fourier transform of the same function defined on R n+2. This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function tf(|t|) and the two-dimensional function (x 1,x 2)↦f(|(x 1,x 2)|). We prove analogous results for radial tempered distributions.

Keywords

Radial Fourier transform Hankel transform 

Mathematics Subject Classification (2000)

42B10 42A10 42B37 

Notes

Acknowledgements

The authors thank Tony Carbery, Hans Georg Feichtinger, Tom H. Koornwinder, Michael Kunzinger, Elijah Liflyand, Michael Oberguggenberger, Norbert Ortner, and Andreas Seeger for helpful discussions and hints with respect to the literature.

References

  1. 1.
    Coifman, R.R., Weiss, G.: Some examples of transference methods in harmonic analysis. In: Symposia Mathematica, vol. XXII, Convegno sull’ Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976, pp. 33–45. Academic Press, London (1977) Google Scholar
  2. 2.
    Evans, L.C.: Partial Differential Equations, 2nd. edn., Graduate Studies in Math., vol. 19. Am. Math. Soc., Providence (2010) zbMATHGoogle Scholar
  3. 3.
    Grafakos, L.: Classical Fourier Analysis, 2nd edn., Graduate Texts in Math., vol. 249. Springer, New York (2008) zbMATHGoogle Scholar
  4. 4.
    Kostenko, A., Sakhnovich, A., Teschl, G.: Weyl–Titchmarsh theory for Schrödinger operators with strongly singular potentials. Int. Math. Res. Not. 2012, 1699–1747 (2012) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Liflyand, E., Trebels, W.: On asymptotics for a class of radial Fourier transforms. Z. Anal. Anwend. 17, 103–114 (1998) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Singh, O.P., Pandey, J.N.: The Fourier–Bessel series representation of the pseudo-differential operator (−x −1 D)ν. Proc. Am. Math. Soc. 115, 969–976 (1992) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975) zbMATHGoogle Scholar
  8. 8.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, New York (1993) zbMATHGoogle Scholar
  9. 9.
    Schaback, R., Wu, Z.: Operators on radial functions. J. Comput. Appl. Math. 73, 257–270 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Stein, E.M., Weiss, G.: Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 31. Princeton University Press, Princeton (1971) zbMATHGoogle Scholar
  11. 11.
    Szmydt, Z.: On homogeneous rotation invariant distributions and the Laplace operator. Ann. Pol. Math. 36, 249–259 (1979) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Teschl, G.: Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Graduate Studies in Math., vol. 99. Am. Math. Soc, Providence (2009) zbMATHGoogle Scholar
  13. 13.
    Treves, J.F.: Lectures on linear partial differential equations with constant coefficients. Notas de Matemática, no. 27. IMPA, Rio de Janeiro (1961) Google Scholar
  14. 14.
    Whitney, H.: Differentiable even functions. Duke Math. J. 10, 159–160 (1943) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Zemanian, A.H.: A distributional Hankel transform. J. SIAM Appl. Math. 14, 561–576 (1966) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Zemanian, A.H.: Generalized Integral Transformations. Interscience, New York (1968) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Faculty of MathematicsUniversity of ViennaWienAustria
  3. 3.International Erwin Schrödinger Institute for Mathematical PhysicsWienAustria

Personalised recommendations