# Definition:Schwartz Test Function

## Definition

Let $\phi : \R \to \C$ be a complex-valued function.

Let $\phi \in \map {\CC^\infty} \R$ be smooth.

Suppose $\phi$ is bounded in the following way:

$\ds \forall m, l \in \N : \sup_{x \mathop \in \R} \size {x^l \map {\phi^{\paren m}} x} < \infty$

where $\phi^{\paren m}$ denotes the $m$-th derivative of $\phi$.

Then $\phi$ is known as a Schwartz test function.

## Source of Name

This entry was named for Laurent-Moïse Schwartz.