# Radial function

In mathematics, a **radial function** is a function defined on a Euclidean space **R**^{n} whose value at each point depends only on the distance between that point and the origin. For example, a radial function Φ in two dimensions has the form

where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and indeed any decent function 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. That is, *ƒ* is radial if and only if

for all ρ ∈ SO(*n*), the special orthogonal group in *n* dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions *S* on **R**^{n} such that

for every test function φ and rotation ρ.

Given any (locally integrable) function *ƒ*, its radial part is given by averaging over spheres centered at the origin. To wit,

where ω_{n−1} is the surface area of the (*n*−1)-sphere *S*^{n−1}, and *r* = |*x*|, *x*′ = *x*/r. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every *r*.

The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. 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.

## See also

## References

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