On Fourier transforms of radial functions and distributions

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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.