# Radial function

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{{short description|Real function on a Euclidean space whose value depends only on distance from the origin}}

In [mathematics](/source/mathematics), a '''radial function''' is a [real-valued function](/source/real-valued_function) defined on a [Euclidean space](/source/Euclidean_space) {{tmath|\R^n}} whose value at each point depends only on the distance between that point and the [origin](/source/Origin_(mathematics)). The distance is usually the [Euclidean distance](/source/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](/source/spherical_harmonics), and any descent function (e.g., [continuous](/source/continuous_function) and [rapidly decreasing](/source/rapidly_decreasing)) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the [solid spherical harmonic](/source/solid_spherical_harmonic) expansion.

A function is radial [if and only if](/source/if_and_only_if) it is invariant under all [rotation](/source/rotation)s 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|''&rho;'' &isin; SO(''n'')}}, the [special orthogonal group](/source/special_orthogonal_group) in {{mvar|n}} dimensions.  This characterization of radial functions makes it possible also to define radial [distributions](/source/distribution_(mathematics)).  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''&minus;1</sub>}} is the surface area of the [(''n''&minus;1)-sphere](/source/N_sphere) {{math|''S''<sup>''n''&minus;1</sup>}}, and {{math|1=''r'' = {{abs|''x''}}}}, {{math|1=''x''&prime; = ''x''/''r''}}.  It follows essentially by [Fubini's theorem](/source/Fubini's_theorem) that a locally integrable function has a well-defined radial part at [almost every](/source/almost_every) {{mvar|r}}.

The [Fourier transform](/source/Fourier_transform) of a radial function is also radial, and so radial functions play a vital role in [Fourier analysis](/source/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>&minus;(''n''&minus;1)/2</sup>}}.  The [Bessel functions](/source/Bessel_functions) are a special class of radial function that arise naturally in Fourier analysis as the radial [eigenfunction](/source/eigenfunction)s of the [Laplacian](/source/Laplacian); as such they appear naturally as the radial portion of the Fourier transform.

==See also==
* [Radial basis function](/source/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

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Adapted from the Wikipedia article [Radial function](https://en.wikipedia.org/wiki/Radial_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Radial_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.