On convolution theorems

Abstract

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let and be two functions with convolution . (Note that the asterisk denotes convolution in this context, and not multiplication. The tensor product symbol is sometimes used instead.) Let denote the Fourier transform operator, so and are the Fourier transforms of and , respectively. Then