On Fourier transforms of radial functions and distributions

TL;DR

This paper derives a formula linking the Fourier transforms of radial functions across different dimensions, facilitating explicit calculations in any dimension based on lower-dimensional transforms.

Contribution

It introduces a new formula connecting Fourier transforms of radial functions in various dimensions, including for distributions, enhancing computational methods.

Findings

Explicit formula relating Fourier transforms in different dimensions

Applicable to radial functions and distributions

Enables calculation of higher-dimensional transforms from lower-dimensional data

Abstract

We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. 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 $t\to f(|t|)$ and the two-dimensional function $(x_1,x_2)\to f(|(x_1,x_2)|)$. We prove analogous results for radial tempered distributions.