Definition:Convolution Integral/Positive Real Domain

Definition

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

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:

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

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.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $2$: The Inverse Laplace Transform: Solved Problems: The Convolution Theorem: $20$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convolution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convolution