Definition:Tempered Distribution

Definition
Let $\map \SS \R$ be a Schwartz space.

Let $\phi, \psi \in \map \SS \R$ be Schwartz test functions.

Let $\alpha \in \C$ be a complex number.

Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \SS \R$ be a convergent sequence with the limit $\mathbf 0 \in \map \SS \R$.

Suppose a mapping $T : \map \SS \R \to \C$ is linear and continuous:


 * $\forall \psi, \phi \in \map \SS \R : \map T {\phi + \psi} = \map T \phi + \map T \psi$


 * $\forall \phi \in \map \SS \R : \forall \alpha \in \C : \map T {\alpha \cdot \phi} = \alpha \cdot \map T \phi$


 * $\paren {\phi_n \stackrel \SS {\longrightarrow} \mathbf 0} \implies \paren {\map T {\phi_n} \to \map T {\mathbf 0}}$

Then $T$ is a tempered distribution.

Also denoted as
The mapping $\map T \phi$ can sometimes be written as $\innerprod T \phi$.

Notation
To avoid confusion between a tempered distribution and the function involved, one can write the function as a subscript.

Suppose we have a tempered distribution $T_f : \map \SS \R \to \C$ and a Schwartz test function $\phi \in \map \SS \R$.

This would correspond to the following mapping:


 * $\ds \phi \stackrel {T_f} {\longrightarrow} \int_{-\infty}^\infty \map f x \map \phi x \rd x$