Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.

Consider two functions ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$ with Fourier transforms ${\displaystyle U}$ and ${\displaystyle V}$:

where ${\displaystyle {\mathcal {F}}}$ denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically ${\displaystyle 2\pi }$ or ${\displaystyle {\sqrt {2\pi }}}$) will appear in the convolution theorem below. The convolution of ${\displaystyle u}$ and ${\displaystyle v}$ is defined by:

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol ${\displaystyle \otimes }$ is sometimes used instead.

The convolution theorem states that:

Applying the inverse Fourier transform ${\displaystyle {\mathcal {F}}^{-1},}$ produces the corollary:

The theorem also generally applies to multi-dimensional functions.

This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.