{{short description|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 {{tmath|\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 {{math|Φ}} in two dimensions has the form<ref>{{Cite news |date=2022-03-17 |title=Radial Basis Function - Machine Learning Concepts |language=en-US |work=Machine Learning Concepts - |url=https://ml-concepts.com/2022/03/17/radial-basis-function/ |access-date=2022-12-23}}</ref> <math display=block>\Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2}</math> where {{mvar|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, {{mvar|f}} is radial if and only if <math display=block>f\circ \rho = f\,</math> for all {{math|''ρ'' ∈ SO(''n'')}}, the special orthogonal group in {{mvar|n}} dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions {{mvar|S}} on {{tmath|\R^n}} such that <math display=block>S[\varphi] = S[\varphi\circ\rho]</math> for every test function {{mvar|φ}} and rotation {{mvar|ρ}}.
Given any (locally integrable) function {{mvar|f}}, its radial part is given by averaging over spheres centered at the origin. To wit, <math display=block>\phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx'</math> where {{math|ω<sub>''n''−1</sub>}} is the surface area of the (''n''−1)-sphere {{math|''S''<sup>''n''−1</sup>}}, and {{math|1=''r'' = {{abs|''x''}}}}, {{math|1=''x''′ = ''x''/''r''}}. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every {{mvar|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 {{math|''R''<sup>−(''n''−1)/2</sup>}}. 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== {{Reflist}} *{{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.|url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.
{{DEFAULTSORT:Radial Function}} Category:Harmonic analysis Category:Rotational symmetry Category:Types of functions