convolution

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convolution

Anatomy any of the numerous convex folds or ridges of the surface of the brain

convolution

(kon-vŏ-loo -shŏn) A mathematical operation that is performed on two functions and expresses how the shape of one is ‘smeared’ by the other. Mathematically, the convolution of the functions f(x) and g(x) is given by
(u )g(x u )du

It finds wide application in physics; it describes, for example, how the transfer function of an instrument affects the response to an input signal. See also autocorrelation function; radio-source structure.

Convolution

 

The convolution of the two functions f1(x) and f2(x) is the function

The convolution of f1(x) and f2(x) is sometimes denoted by f1 * f2

If f1 and f2 are the probability density functions of two independent random variables X and Y, then f1 * f2 is the probability density function of the random variable X + Y. If Fk(x) is the Fourier transform of the function fk(x), that is,

then F1(x) F2(x) is the Fourier transform of the function f1 * f2. This property of convolutions has important applications in probability theory. The convolution of two functions exhibits an analogous property with respect to the Laplace transform; this fact underlies broad applications of convolutions in operational calculus.

The operation of convolution of functions is commutative and associative—that is, f1 * f2 = f2 * f1 and f1 * (f2 * f3) = (f1 * f2) * f3. For this reason, the convolution of two functions can be regarded as a type of multiplication. Consequently, the theory of normed rings can be applied to the study of convolutions of functions.

convolution

[‚kän·və′lü·shən]
(anatomy)
A fold, twist, or coil of any organ, especially any one of the prominent convex parts of the brain, separated from each other by depressions or sulci.
(geology)
The process of developing convolute bedding.
A structure resulting from a convolution process, such as a small-scale but intricate fold.
(mathematics)
The convolution of the functions ƒ and g is the function F, defined by
(statistics)
A method for finding the distribution of the sum of two or more random variables; computed by direct integration or summation as contrasted with, for example, the method of characteristic functions.
References in periodicals archive ?
A rather difficult method for calculating the convolution integrals based on representations of the factors by step-functions, was given by I.
Raina, A class of convolution integral equations, J.
Moreover, a fine subdivision of the grid for the sea cells adjacent to the observation point is needed to provide better accuracy in the convolution integral.
For the sake of simplicity of presentation and implementation, we have invoked the assumption of time invariance; this assumption is what leads to the specific convolution integral given in Equation 7, and repeated below:
We therefore compute the cross correlation between Y(t) and X(t) using the definition of convolution integral as where p(a,t) and q(a,t) represent the results of cross correlation obtained after carrying out integration of eq.
In contrast with CWR-BUCKLE program, which use probability density functions, the probabilistic computational algorithm of SCFJ program is based on the evaluation of convolution integrals (Ghiocel & Lungu, 1975) in a discrete approach (Ghiocel & Lungu, 1982), using the histograms of the main parameters which characterize the stability of the CWR track (Ungureanu & Dosa, 2007).
Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals.