“Radial function”的意思、由来-开放百科全书

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Radial function

释义

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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 functions, and any decent 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, ƒ is radial if and only if

for all {{nowrap|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 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 ω_n−1 is the surface area of the (n−1)-sphere Sⁿ⁻¹, and {{nowrap|1=r = {{abs|x}}}}, {{nowrap|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 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

{{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.}}.

3 : Harmonic analysis|Rotational symmetry|Types of functions

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