Definition:Test Function

Definition

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

Let $K \subseteq \R^d$ be a compact subset.

Let $K$ be the support of $\phi$:

$\map \supp \phi = K$

Let $\phi$ be a smooth function across $K$ with respect to all its variables.

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

Also known as

Test functions are also known as bump functions.

Examples

Exponential of $\dfrac 1 {x^2 - 1}$

The graph of the test function. It is smooth everywhere. Outside of its support denoted by dashed lines the function is identically zero. At boundary points it connects smoothly.

Let $\phi : \R \to \R$ be a real function with support on $x \in \closedint {-1} 1$ such that:

$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$

Then $\phi$ is a test function.

Also see

• Results about test functions can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples