The Fourier transform of multiradial functions

TL;DR

This paper derives an exact formula for the Fourier transform of multiradial functions, which depend on the norms of multiple vector components, expressed through the Fourier transform of a related function in lower-dimensional spaces.

Contribution

It provides a novel explicit formula for the Fourier transform of multiradial functions in terms of simpler Fourier transforms, expanding analytical tools for such functions.

Findings

Exact Fourier transform formula for multiradial functions

Expresses transform in terms of lower-dimensional Fourier transforms

Facilitates analysis of multiradial functions in harmonic analysis

Abstract

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form $\Phi(x)=\phi(|x_1|, \dots, |x_m|)$, $x_i\in \mathbf R^{n_i}$, in terms of the Fourier transform of the function $\phi$ on $\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}$, where $r_i$ is either 1 or 2.