Convolution theorem

The convolution theorem states that the Fourier transform of a convolution is the point-wise product of Fourier transforms.

Let f and g be two functions with convolution f * g. (Note that the asterisk denotes convolution in this context, and not multiplication.) Let F denote the Fourier transform operator, so F f and F g are the Fourier transforms of f and g, respectively. Then

F (f * g) = (F f) · (F g),

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

F (f · g) = (F f) * (F g).

By applying 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.