Radial function

{{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{{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}}

$$\backslash Phi(x,y)\; =\; \backslash varphi(r),\; \backslash quad\; r\; =\; \backslash sqrt\{x^2+y^2\}$$

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

$$f\backslash circ\; \backslash rho\; =\; f\backslash ,$$

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

$$S[\backslash varphi]\; =\; S[\backslash varphi\backslash circ\backslash rho]$$

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,

$$\backslash phi(x)\; =\; \backslash frac\{1\}\{\backslash omega\_\{n-1\}\}\backslash int\_\{S^\{n-1\}\}\; f(rx\text{'})\backslash ,dx\text{'}$$

where {{math|ω_n−1}} is the surface area of the (n−1)-sphere {{math|Sⁿ⁻¹}}, 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^−(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.