Transforms and Convolutions

There are many theorems of the form

where f and g are functions, T is an integral transform, and * is a kind of convolution. In words, the transform of a convolution is the product of transforms.

When the transformation changes, the notion of convolution changes.

Here are three examples.

Fourier transform and convolution

With the Fourier transform defined as

convolution is defined as

Note: There are many minor variations on the definition of the Fourier transform. See these notes.

Laplace transform and convolution

With the Laplace transform defined as

convolution is defined as

Mellin transform and convolution

With the Mellin transform defined as

convolution is defined as

Related posts

• Unified theory of Fourier transforms
• Relating Fourier series and Fourier transforms
• Generalized Fourier transforms
• Outline of Laplace transforms