On Fourier Transforms of Radial Functions and Distributions

• Published in: The Journal of fourier analysis and applications Vol. 19; no. 1; pp. 167 - 179
• Main Authors: Grafakos, Loukas, Teschl, Gerald
• Format: Journal Article
• Language: English
• Published: Boston SP Birkhäuser Verlag Boston 01-02-2013; Springer
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Signal,Image and Speech Processing; Hankel transform; Radial Fourier transform; 42B37; 42A10; 42B10
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-012-9242-5

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.