Radial Function

A Radial Function is a vector function defined on an Euclidean space [math]\displaystyle{ \R^n }[/math] whose value at each point depends only on the distance between that point and the origin.

• Example(s):
  • a 2-D Radial Function, [math]\displaystyle{ \Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2} }[/math]
  • a Radial Basis Function.
  • …
• Counter-Example(s):
  • a Linear Function.
• See: Spherical Function, Solid Spherical Harmonic, Rotation, N Sphere, Fubini's Theorem, Fourier Transform.

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References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/radial_function Retrieved:2014-9-22.
  • In mathematics, a 'radial function is a function defined on an Euclidean space Rn 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  :[math]\displaystyle{ \Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2} }[/math]

    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  :[math]\displaystyle{ f\circ \rho = f\, }[/math]

    for all, 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 Rn such that  :[math]\displaystyle{ S[\phi] = S[\varphi\circ\rho] }[/math]

    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,  :[math]\displaystyle{ \phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx' }[/math]

    where ω_n−1 is the surface area of the (n−1)-sphere Sⁿ⁻¹, and , . 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.

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