Radial function

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 in two dimensions has the form \Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2} where 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, is radial if and only if f\circ \rho = f\, for all , the special orthogonal group in dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions on such that S[\varphi] = S[\varphi\circ\rho] for every test function and rotation . Given any (locally integrable) function , 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 is the surface area of the (n−1)-sphere , and , .…

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 in two dimensions has the form \Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2} where 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, is radial if and only if f\circ \rho = f\, for all , the special orthogonal group in dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions on such that S[\varphi] = S[\varphi\circ\rho] for every test function and rotation . Given any (locally integrable) function , 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 is the surface area of the (n−1)-sphere , and , . It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every . 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 . 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 ==References== .

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