Radial function

In mathematics, a radial function is a real-valued function defined on a Euclidean space Template:Tmath 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 Template:Math in two dimensions has the form $$\mathrm{\Phi }(x,y)=\phi (r),r=\sqrt{x^2+y^2}$$ where Template:Mvar 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, Template:Mvar is radial if and only if $$f\circ \rho =f$$ for all Template:Math, the special orthogonal group in Template:Mvar dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions Template:Mvar on Template:Tmath such that $$S[\phi ]=S[\phi \circ \rho ]$$ for every test function Template:Mvar and rotation Template:Mvar.

Given any (locally integrable) function Template:Mvar, its radial part is given by averaging over spheres centered at the origin. To wit, $$\varphi (x)=\frac{1}{{\omega }_{n-1}}\underset{S^{n-1}}{\int }f(r{x}^{\prime })d{x}^{\prime }$$ where Template:Math is the surface area of the (n−1)-sphere Template:Math, and Template:Math, Template:Math. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every Template:Mvar.

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 Template:Math. 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

• Radial basis function

References

• Template:Citation.