# Definition:Convolution Integral

This page is about Convolution Integral in the context of Integral Calculus. For other uses, see Convolution.

## Definition

Let $f$ and $g$ be real functions which are integrable.

The convolution integral of $f$ and $g$ is defined as:

$\ds \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$

### Positive Real Domain

Let $f$ and $g$ be supported on the positive real numbers $\R_{\ge 0}$ only.

The convolution integral of $f$ and $g$ may be defined as:

$\ds \map f t * \map g t := \int_0^t \map f u \map g {t - u} \rd u$

### Cross-Correlation

The cross-correlation of $f$ and $g$ is defined as:

$\ds \map f t \star \map g t := \int_{-\infty}^\infty \map f u \map g {t + u} \rd u$

## Also known as

A convolution integral is also generally known just as a convolution.

However, it has to be borne in mind that the term convolution in general has a wider definition.

## Also see

• Results about convolution integrals can be found here.