Radial function

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title: "Radial function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["harmonic-analysis", "rotational-symmetry", "types-of-functions"] description: "Real function on a Euclidean space whose value depends only on distance from the origin" topic_path: "general/harmonic-analysis" source: "https://en.wikipedia.org/wiki/Radial_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 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 \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\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 such that 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, \phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx'),dx' where ωn−1 is the surface area of the (n−1)-sphere S**n−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.

References

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References

1. (2022-03-17). "Radial Basis Function - Machine Learning Concepts". Machine Learning Concepts -.

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