Linearizations via Dirac delta function

This note is to show that the position-space embedding in \cite{ESP2021embedding} in the position and occupation bases can be obtained by considering the dynamics of Dirac delta function $$\delta (\mathbf{x}-\mathbf{z}(t))=\delta (x_1-z_1(t))\cdots \delta (x_d-z_d(t)),$$ where $\mathbf{z}(t)\in {\mathbb{R}}^d$ is the solution of a nonlinear dynamical system and $\mathbf{x}\in {\mathbb{R}}^d$ is a variable in the position space.