Definition (3.3.2)

A distribution is a linear, continuous functional on the space of test functions (aka Schwartz functions) $\mathcal{S}$.

Definition (Regular distribution)

A regular distribution, $\bar{\gamma}\in \mathcal{S}^{*}$, is a specific form of distribution that is expressed in the following form: $$\bar{\gamma}(\phi):=\int\limits _{\mathbb{R}}\gamma(t)\phi(t) \, dt ,\quad\phi \in \mathcal{S}$$where $\bar{\gamma}$ is a regular distribution on $\mathcal{S}$ represented by the function $\gamma(t)$.

Definition (3.3.3)

A tempered function, $x(t)$ is one which satisfies small growth, that is, for some $\beta,\gamma \in \mathbb{R},N\in \mathbb{Z}_{+}$: $$|x(t)|\le \beta|t|^{N}+\gamma,\quad\forall t\in \mathbb{R}$$

Definition (3.3.4)

A sequence of distributions $\{ \bar{\gamma}_{n} \}_{n\in \mathbb{N}}$ converges to a distribution $\bar{\gamma}$ if $\bar{\gamma}_{n}$ converges in the weak* sense to $\bar{\gamma}$.