Radial function

In mathematics, a radial function is a real-valued function defined on a Euclidean space ⁠ R n {\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 Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 {\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 f ∘ ρ = f {\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 ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ such that S [ φ ] = S [ φ ∘ ρ ] {\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, ϕ ( x ) = 1 ω n − 1 ∫ S n − 1 f ( r x ′ ) d x ′ {\displaystyle \phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'} where ωn−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.