In functional analysis, convolution, convolution or convolution is a mathematical operation that generates the third function from two functions F and G, and its essence is a special integral transformation, in which the product of the overlapping parts of F and G is inverted and translated to represent the integral of the overlapping length.

If the function involved in convolution is regarded as interval indicator function, convolution can also be regarded as a generalization of "moving average".

Convolution (also called convolution) and deconvolution (also called deconvolution) are a mathematical method of integral transformation, which are widely used in many aspects. Using convolution technology to solve well test interpretation problems has achieved good results.

Convolution theorem

Convolution theorem points out that the Fourier transform of function convolution is the product of function Fourier transform. That is, convolution in one domain is equivalent to product in another domain, for example, convolution in time domain corresponds to product in frequency domain.

This theorem is also applicable to various Fourier transform variants, such as Laplace transform, bilateral Laplace transform, Z transform, Merlin transform and Hartley transform (see Merlin inversion theorem). In harmonic analysis, it can also be extended to Fourier transform defined on locally compact Abelian groups.

Convolution theorem can simplify convolution operation. For a sequence of length n, it needs (2n- 1) bit multiplication according to the definition of convolution, and its computational complexity is as follows: after the sequence is transformed into frequency domain by Fourier transform, only one set of bit multiplication is needed. After using the fast algorithm of Fourier transform, the total computational complexity is. This result can be applied to fast multiplication calculation.