📂Distribution Theory

Regulating Supersaturation

Definition¹

A continuous linear functional $T:\mathcal{S}(\mathbb{R}^{n}) \to \mathbb{C}$ on the Schwartz space is called a tempered distribution. In other words, a tempered distribution is an element of the dual space of the Schwartz space. Therefore,

$$ T \in \mathcal{S}^{ \ast } $$

is denoted as such, and $\mathcal{S}^{ \ast }$ is called the space of tempered distributions.

Description

Since a tempered distribution $T$ is linear, the following holds true.

$$ T(a\phi + b \psi) = aT(\phi) + bT(\psi)\quad \left( \phi,\psi \in \mathcal{S}(\mathbb{R}^{n}),\ a,b\in\mathbb{C} \right) $$

It is also continuous, hence the following is true.

$$ \left\{ \phi _{n} \right\} \to \phi \in \mathcal{S}(\mathbb{R}^{n}) \implies \left\{ T(\phi_{n}) \right\} \to T(\phi) \quad \left( \phi_{n},\phi \in \mathcal{S}(\mathbb{R}^{n})\right) $$

In the case of distributions, the test functions which are elements of the domain, have a compact support, so it did not matter how fast their values increased. However, Schwartz functions, unlike test functions, do not have compact support, so a tempered distribution cannot increase too rapidly. The term tempered is derived from this meaning, and for the same reason, tempered distributions are also called distributions of slow growth.