(redirected from convolutions)
Also found in: Dictionary, Thesaurus, Medical, Legal, Wikipedia.
Related to convolutions: convolving


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


(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.



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.


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.
The process of developing convolute bedding.
A structure resulting from a convolution process, such as a small-scale but intricate fold.
The convolution of the functions ƒ and g is the function F, defined by
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 ?
It is noted that the above convolution formula is reduced to the Hadamard product if the co-analytic part of the functions [f.
y] in the right hand side of (8) and using (26), we obtain the simplified equation without any explicit convolutions and temporal derivatives as:
We now derive a convolution characterization for functions in the class [[summation].
We first introduce some spaces for sine approximation, and then describe sine interpolation, sine indefinite integration and sine convolution.
We excited elastic wave by breaking a pencil lead at the crown top of several convolutions from the flange.
In 1955-1956 Malgrange [14] proved an existence theorem for convolution equations on the Frechet space of entire functions H([C.
CNAPS accelerates up to 100-fold image processing and recognition functions such as filtering and convolutions, and image manipulation functions such as rotation and color transformation.
Finally in Section 5, we derive interesting and useful convolution results with prestarlike functions for functions in these classes.
Written by Clive Exton and directed by Tom Clegg, ``Murder in Mesopotamia'' plays out cleanly and clearly despite the narrative convolutions, and boasts handsome visual design.
Adaptive's CNAPS accelerates up to 100-fold image processing and recognition functions such as character recognition, filtering and convolutions and image manipulation functions such as rotation and resizing.
Morphing between a sort of sci-fi behemoth and fantastic hot rod, the work comprised everything from sections of surrounding hillside to a river, with the baroque convolutions of an elegant water-circulation system begging for scrupulous examination.
Screenwriters Robert Palm, Wendy West and Richard Sweren (working off a story by Palm, Rene Balcer and franchise creator Dick Wolf) keep the narrative convolutions fascinating, sensational and miraculously straightforward.