Radial function

Inmathematics,aradial functionis a real-valuedfunctiondefined on aEuclidean spaceRⁿwhose value at each point depends only on the distance between that point and the origin. The distance is usually theEuclidean distance.For example, a radial function Φ in two dimensions has the form

$\Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}$

where φ is a function of a single non-negative real variable. Radial functions are contrasted withspherical functions,and any descent function (e.g.,continuousandrapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: thesolid spherical harmonicexpansion.

A function is radialif and only ifit is invariant under allrotationsleaving the origin fixed. That is,ƒis radial if and only if

$f\circ \rho =f\,$

for allρ ∈ SO(n),thespecial orthogonal groupinndimensions. This characterization of radial functions makes it possible also to define radialdistributions.These are distributionsSonRⁿsuch that

$S[\varphi ]=S[\varphi \circ \rho ]$

for every test function φ and rotation ρ.

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

$\phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'$

where ω_n−1is the surface area of the(n−1)-sphereSⁿ⁻¹,andr= |x|,x′ =x/r.It follows essentially byFubini's theoremthat a locally integrable function has a well-defined radial part atalmost everyr.

TheFourier transformof a radial function is also radial, and so radial functions play a vital role inFourier 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 thanR^−(n−1)/2.TheBessel functionsare a special class of radial function that arise naturally in Fourier analysis as the radialeigenfunctionsof theLaplacian;as such they appear naturally as the radial portion of the Fourier transform.

See also

• Radial basis function

References

• Stein, Elias;Weiss, Guido(1971),Introduction to Fourier Analysis on Euclidean Spaces,Princeton, N.J.: Princeton University Press,ISBN978-0-691-08078-9.