# 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*)d

*u*

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 *f*_{1}(*x*) and *f*_{2}(*x*) is the function

The convolution of *f*_{1}(*x*) and *f*_{2}(*x*) is sometimes denoted by *f*_{1} * *f*_{2}

If *f*_{1} and *f*_{2} are the probability density functions of two independent random variables *X* and *Y*, then *f*_{1} * *f*_{2} is the probability density function of the random variable *X* + *Y*. If *F _{k}*(

*x*) is the Fourier transform of the function

*f*(

_{k}*x*), that is,

then *F*_{1}(*x*) *F*_{2}(*x*) is the Fourier transform of the function *f*_{1} * *f*_{2}. 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, *f*_{1} * *f*_{2} = *f*_{2} * *f*_{1} and *f*_{1} * (*f*_{2} * *f*_{3}) = (*f*_{1} * *f*_{2}) * *f*_{3}. 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]*g*is the function

*F,*defined by