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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.

F(g(x)*f(x)) = F(g(x))F(f(x))

Where f stands for Fourier transform. 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, 2n- 1 set of contraposition multiplication is needed according to the definition of convolution, and its computational complexity is: after the sequence is transformed into frequency domain by Fourier transform, only one set of contraposition 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.

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