${\ensuremath{K}}$ using the commutative diagrams in Figure \[fig:commute\_sysop\_dyn\] and Figure \[fig:commute\_sysop\_meas\] respectively.

The finite-dimensional evolution equations equivalent to [(\[eq:ideal\_lin\_evol\])]{} in the dual space can be formulated as $$\begin{aligned}
  {w}_{k+1} &= A{w}_k + \eta_k {\label{eq:k_measure}}\\
 y_{k} &= {\ensuremath{K}}_{k} w_{k} +\zeta_k, {\label{eq:k_measure1}}\end{aligned}$$ where we have matrices $A\in {\mathbb{R}}^{{M}\times{M}}, \ {\ensuremath{K}}_{k}\in {\mathbb{R}}^{{N}\times{M}}$, the vectors ${w}_k, {w}\in{\mathbb{R}}^{{M}}$, and we have slightly abused notation to let $\eta_k$ and $\zeta_k$ denote their ${\ensuremath{{\mathcal{H}}_{{C}}}}$ counterparts. Note that the measurement operator ${\mathcal{K}}$ is simply a sampling of the function $f$ at an arbitrary set of sensing locations ${\mathcal{X}}= {\{x_1, \dots, x_{{N}}\}}$, where $x_i\in{\Omega}$: we will see how this affects the structure of ${\ensuremath{K}}_k$ momentarily. The equations [(\[eq:ideal\_lin\_evol\])]{} suggest an immediate extension to functional control problems. Pick another dictionary of atoms ${\mathcal{F}_{{D}}}= \begin{bmatrix}{\psi}(d_1) &\cdots & {\psi}(d_{{\ell}})
\end{bmatrix}$, ${\psi}(d_j)\in{\mathcal{H}}$, $d_j\in{\Omega}$, the span of which, denoted by ${\ensuremath{{\mathcal{H}}_{{D}}}}$, is a strict subspace of the RKHS ${\mathcal{H}}$ generated by the kernel. The functional evolution equation is then as follows: $$\begin{aligned}
{\label{eq:ideal_lin_evol_control}}
 f_{k+1} &= {\mathcal{A}}f_k + {\mathcal{B}}{\delta}_k + \eta_k\\
 y_k &= {\mathcal{K}}_k f_k + \zeta_k,\end{aligned}$$ where the control functions ${\delta}_k$ evolve in ${\ensuremath{{\mathcal{H}}_{{D}}}}$, and ${\mathcal{B}}:{\ensuremath{{\mathcal{H}}_{{D}}}}\to{\ensuremath{{\mathcal{H}}_{{C}}}}$. To derive the finite-dimensional equivalent of ${\mathcal{B}}$, we have to work out the structure of the matrix $B$: 