On Fourier transforms of radial functions and distributions

2011/12/31 by Loukas Grafakos, Gerald Teschl · 1 citation
Mathematics · Physics and Astronomy · #math.CA #math-ph #math.AP #math.MP #msc:42B10 #msc:42A10 #msc:42B37

paper · pdf · doi:10.1007/s00041-012-9242-5

published as J. Fourier Anal. Appl. 19, 167-179 (2013) · 12 pages

arxiv created 2013/02/18 · arxiv updated 2013/02/19

Abstract

We find a formula that relates the Fourier transform of a radial function on Rⁿ with the Fourier transform of the same function defined on Rⁿ⁺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 (x₁,x₂)→ f(|(x₁,x₂)|). We prove analogous results for radial tempered distributions.

Cited by

• Fourier transform [wikipedia]

Related

• Slicing of radial functions: a dimension walk in the Fourier space
• The Fourier transform of multiradial functions
• The kernel of the radially deformed Fourier transform
• Some Conditions for Existence and Integrability of the Fourier Transform
• A Fourier transform for all generalized functions