Radial function
Encyclopedia
In mathematics
, a radial function is a function
defined on a Euclidean space
Rn whose value at each point depends only on the distance between that point and the origin. For example, a radial function Φ in two dimensions has the form
where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical function
s, and indeed any decent function 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 rotation
s leaving the origin fixed. That is, ƒ is radial if and only if
for all , 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 Rn such that
for every test function φ and rotation ρ.
Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit,
where ωn−1 is the surface area of the (n−1)-sphere Sn−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 eigenfunction
s of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a radial function is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
defined on a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn whose value at each point depends only on the distance between that point and the origin. For example, a radial function Φ in two dimensions has the form
where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical function
Spherical function
Spherical function can refer to* Spherical harmonics*Zonal spherical function...
s, and indeed any decent function 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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is invariant under all rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s leaving the origin fixed. That is, ƒ is radial if and only if
for all , the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
. These are distributions S on Rn such that
for every test function φ and rotation ρ.
Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit,
where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and , . It follows essentially by Fubini's theorem
Fubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
that a locally integrable function has a well-defined radial part at almost every r.
The Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
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 eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.