# 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

## 2 thoughts on “Transforms and Convolutions”

1. While you’re at it, you might well add moment generating functions for probability distributions.

On the discrete side there is the z-transform and the probability generating function.

2. Raul Hindov

Another similar example is Dirichlet convolution and Dirichlet series for arithmetic functions.