the dual space of ${\mathcal{H}}$. For concreteness, we work with an approximate space as follows: given points ${\mathcal{C}}= {\{c_1,\dots,c_{{M}}\}}$, $c_i\in{\Omega}$, we have a dictionary of atoms ${\mathcal{F}_{{C}}}= \begin{bmatrix}{\psi}(c_1) &\cdots & {\psi}(c_{{M}})
\end{bmatrix}$, ${\psi}(c_i)\in{\mathcal{H}}$, the span of which is a strict subspace of the RKHS generated by the kernel. Formally, we have $$\begin{aligned}
 {\mathcal{C}}\mapsto {\ensuremath{{\mathcal{H}}_{{C}}}}:= \operatorname{span}\begin{bmatrix}{\psi}(c_1) &\cdots & {\psi}(c_{{M}})
\end{bmatrix}
\subset {\mathcal{H}}.\end{aligned}$$ This regime, which trades off the flexibility of a truly nonparametric approach for computational realizability, still allows for the representation of rich phenomena. Let ${N}$ represent the number of sampling locations, and ${M}$ be the number of bases generating ${\ensuremath{{\mathcal{H}}_{{C}}}}$. Note that every function $f\in{\ensuremath{{\mathcal{H}}_{{C}}}}$ has an expansion of the form $$\begin{aligned}
{\label{eq:expansion}}
 f(x) = \sum_{i=1}^{{M}} w_i k(c_i,x).\end{aligned}$$ This expansion allows us to write the $w_i$ coordinates in the dual space as vectors $w\in{\mathbb{R}}^{{M}}$. We can show the relation of the function spaces to their Euclidean counterparts via commutative diagrams. Define ${\ensuremath{\mathcal{W}}}:{\ensuremath{{\mathcal{H}}_{{C}}}}\to{\mathbb{R}}^{{M}}$ as the operator that maps the coordinates $w_i$ in [(\[eq:expansion\])]{} to vectors ${w}\in{\mathbb{R}}^{{M}}$, and let ${\ensuremath{\mathcal{W}^{-1}}}:{\mathbb{R}}^{{M}}\to{\ensuremath{{\mathcal{H}}_{{C}}}}$. Note that for finite-dimensional spaces, this inverse map always exists. These definitions allow us to outline the relations between the dynamics operators ${\mathcal{A}}$ and $A$, and the measurement operators ${\mathcal{K}}$ and 