Convolution Theorem

The convolution theorem is a result in Fourier analysis stating that, under suitable conditions, the Fourier transform of the convolution of two functions equals the ordinary product of their individual Fourier transforms. Equivalently, an operation called convolution performed in one domain, such as time, corresponds to simple pointwise multiplication in the corresponding transformed domain, such as frequency. The theorem has counterparts for other transforms related to the Fourier transform, and it is what makes it possible, in many applications, to replace a computationally expensive convolution with a much simpler multiplication after transforming the functions involved.