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


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.


Radial Fourier transform Hankel transform 

Mathematics Subject Classification (2000)

42B10 42A10 42B37 



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.


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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

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