# Definition:Schwartz Test Function

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## Definition

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

Let $\phi \in \map {C^\infty} \R$ 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**.

## Also see

- Results about
**Schwartz test functions**can be found**here**.

## Source of Name

This entry was named for Laurent-Moïse Schwartz.

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 6.5$: A glimpse of distribution theory. Fourier transform of (tempered) distributions