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On Fourier Transforms of Radial Functions and Distributions

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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 t↦f(|t|) and the two-dimensional function (x 1,x 2)↦f(|(x 1,x 2)|). We prove analogous results for radial tempered distributions.

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

  2. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Gerald Teschl

  3. International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Wien, Austria

    Gerald Teschl

Authors
  1. Loukas Grafakos
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  2. Gerald Teschl
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Corresponding author

Correspondence to Loukas Grafakos.

Additional information

Communicated by Arieh Iserle.

Grafakos’ research was supported by the NSF (USA) under grant DMS 0900946. Teschl’s work was supported by the Austrian Science Fund (FWF) under Grant No. Y330.

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Grafakos, L., Teschl, G. On Fourier Transforms of Radial Functions and Distributions. J Fourier Anal Appl 19, 167–179 (2013). https://doi.org/10.1007/s00041-012-9242-5

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  • DOI: https://doi.org/10.1007/s00041-012-9242-5

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