Physlib.Mathematics.Distribution.PowMul

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$) and let $f: \mathbb{R} \to \mathbb{k}$ be the canonical continuous linear embedding defined by $f(x) = x$. For any $x \in \mathbb{R}$ and any $i \in \mathbb{N}$, the norm of the $i$-th iterated Fréchet derivative of $f$ at $x$ is given by: $$\left\| D^i f(x) \right\| = \begin{cases} |x| & \text{if } i = 0 \\ 1 & \text{if } i = 1 \\ 0 & \text{if } i \geq 2 \end{cases}$$

The continuous linear map $L: \mathcal{S}(\mathbb{R}, \mathbb{k}) \to \mathcal{S}(\mathbb{R}, \mathbb{k})$ defined by mapping a Schwartz function $\psi$ to the function $x \mapsto x \psi(x)$, where $\mathcal{S}(\mathbb{R}, \mathbb{k})$ is the Schwartz space of rapidly decreasing functions on $\mathbb{R}$ taking values in the field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$).

Let $\mathcal{S}(\mathbb{R}, \mathbb{k})$ denote the Schwartz space of rapidly decreasing functions from $\mathbb{R}$ to a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{k})$ and any $x \in \mathbb{R}$, the value of the Schwartz function $(\text{powOneMul } \psi)$ evaluated at $x$ is given by $x \cdot \psi(x)$.