Radial function

In mathematics, a radial function is a real-valued function defined on a Euclidean space ⁠${\displaystyle \mathbb {R} ^{n}}$⁠ whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form^[1] $${\displaystyle \Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}}$$ 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. That is, f is radial if and only if $${\displaystyle f\circ \rho =f\,}$$ 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 ⁠${\displaystyle \mathbb {R} ^{n}}$⁠ such that $${\displaystyle S[\varphi ]=S[\varphi \circ \rho ]}$$ 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. To wit, $${\displaystyle \phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'}$$ where ω_n−1 is the surface area of the (n−1)-sphere Sⁿ⁻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.