For example, a radial function Φ in two dimensions has the form[1]
where φ is a function of a single non-negative real variable.
Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion.
A function is radial if and only if it is invariant under all rotations leaving the origin fixed.
for all ρ ∈ SO(n), the special orthogonal group in n dimensions.
for every test function φ and rotation ρ.
Given any (locally integrable) function f, its radial part is given by averaging over spheres centered at the origin.
Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2.
The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.