Convolution theorem

The convolution theorem states that the convolution is transformed into a point-wise multiplication by the Fourier transform.

Let f and g be two functions with convolution f * g. Let F be the operator performing the Fourier transform such that e.g. F f is the Fourier transform of f. Then

F (f * g) = (F f) · (F g),

where · denotes the point-wise multiplication. It also works "the other way round":

F (f · g) = (F f) * (F g).

By using the inverse Fourier transfrom F^-1, we can write

f * g = F^-1 (F f · F g),

a formulation which is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity. With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced to O(n log n). This can be exploited to construct fast multiplication algorithms.