Radial function (original) (raw)

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Real function on a Euclidean space whose value depends only on distance from the origin

In mathematics, a radial function is a real-valued function defined on a Euclidean space ⁠ R n {\displaystyle \mathbb {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[1] Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 {\displaystyle \Phi (x,y)=\varphi (r),\quad r={\sqrt {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\,} {\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}} {\displaystyle \mathbb {R} ^{n}}⁠ such that S [ φ ] = S [ φ ∘ ρ ] {\displaystyle S[\varphi ]=S[\varphi \circ \rho ]} {\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'} {\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 S _n_−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.

  1. ^ "Radial Basis Function - Machine Learning Concepts". Machine Learning Concepts -. 2022-03-17. Retrieved 2022-12-23.