, Volume 19, Issue 1, pp 167-179

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

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.