Convolution

convolution

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

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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 andgis the functionF,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.

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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 F_k(x) is the Fourier transform of the function f_k(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.