In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn 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[1]
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, ƒ is radial if and only if
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 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 r = |x|, 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.
In mathematics, a radialfunction is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between...
In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input...
mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the...
In statistical mechanics, the radial distribution function, (or pair correlation function) g ( r ) {\displaystyle g(r)} in a system of particles (atoms...
In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In...
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured...
common activation functions can be divided into three categories: ridge functions, radialfunctions and fold functions. An activation function f {\displaystyle...
In mathematics, a radially unbounded function is a function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which ‖ x ‖ → ∞...
to the carpus, radial deviation may be reinforced if forearm muscles are functioning poorly or have abnormal insertions. Although radial longitudinal deficiency...
the integral of a function on Rn is related to the integral of the function over spherical shells: level sets of the radialfunction. The formula plays...
reference to the usual codomain of the arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to...
Bessel function of the first kind with order n + 2k − 2/2. When k = 0 this gives a useful formula for the Fourier transform of a radialfunction. This...
\theta ,\phi )=R(r)\,\,Y_{\ell }^{m}\!(\theta ,\phi )} where R are radialfunctions and Ym ℓ(θ, φ) are spherical harmonics of degree ℓ and order m. This...
for a hydrogen-like atom. The part of the function that depends on distance r from the nucleus has radial nodes and decays as e − α r {\displaystyle...
equation as in the case of the radial wave function. However, the range of the variable is different: in the radial wave function, ξ ≥ 1 {\displaystyle \xi...
A radial flux motor generates flux perpendicular to the axis of rotation. By contrast, an axial flux motor generates flux parallel to the axis. The features...
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's...
Radial tunnel syndrome (RTS) is a compression neuropathy of the radial nerve as it travels from the upper arm (the brachial plexus) to the hand and wrist...
spherical coordinates to show that the solution was a product of a radialfunction P ( r ) / r {\displaystyle P(r)/r} and a spherical harmonic with an...
provides motor function to the muscles in the posterior compartment of the forearm, which is mostly the extensor muscles of the hand. The radial nerve arises...
integer. The radial component R has the form R ( r ) = γ J n ( ρ ) , {\displaystyle R(r)=\gamma J_{n}(\rho ),} where the Bessel function Jn(ρ) satisfies...
Radial glial cells, or radial glial progenitor cells (RGPs), are bipolar-shaped progenitor cells that are responsible for producing all of the neurons...
patterns in the feature space. Normal distribution Cauchy distribution Radial basis function kernel Squires, G. L. (2001-08-30). Practical Physics (4 ed.). Cambridge...
the resulting radial-velocity amplitude allows the calculation of the lower bound on a planet's mass using the binary mass function. Radial velocity methods...
f {\displaystyle f} . Of special practical importance is the radial distribution function, which is independent of orientation. It is a major descriptor...
importance of the radial spoke in the proper function of these organelles, its structure and mode of action remain poorly understood. Radial spokes are T-shaped...
the masses of the components. The binary mass function follows from Kepler's third law when the radial velocity of one binary component is known. Kepler's...
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![{\displaystyle S[\varphi ]=S[\varphi \circ \rho ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8d370d344493bb022d289cb6f04c356e93cb7c)
