|
In mathematics, a radial function is a function defined on a Euclidean space R''n'' 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 , 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 R''n'' 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 ''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. ==See also== * Radial basis function 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「radial function」の詳細全文を読む スポンサード リンク
|