since ${\ensuremath{{\mathcal{H}}_{{C}}}}$ is not, in general, isomorphic to ${\ensuremath{{\mathcal{H}}_{{D}}}}$, this imposes strict restrictions on $B$. We derive $B$ using least squares using the inner product of ${\mathcal{H}}$. Let ${\delta}= \sum_{j=1}^{{\ell}}{\acute{{w}}}_j{k}(d_j,x)$, and let ${\mathcal{F}_{{C}}}= 
 \begin{bmatrix}{\psi}(c_1) &\cdots & {\psi}(c_{{M}})
\end{bmatrix}$ be the basis for ${\ensuremath{{\mathcal{H}}_{{C}}}}$. Then the projection of $\delta$ onto ${\ensuremath{{\mathcal{H}}_{{C}}}}$ can be derived as $$\begin{aligned}
 \begin{bmatrix}
  \l \delta, {\psi}(c_1) \r_{{\mathcal{H}}}\\
   \vdots\\
  \l \delta, {\psi}(c_{{M}}) \r_{{\mathcal{H}}} 
 \end{bmatrix}
 &= 
 \underbrace{
 \begin{bmatrix}
  {k}(d_1,c_1) & \cdots & {k}(d_{{\ell}},c_1)\\
   \vdots  &\ddots &\vdots\\
  {k}(d_1,c_{{M}}) & \cdots & {k}(d_{{\ell}},c_{{M}})
 \end{bmatrix}}_{{{\ensuremath{K}}_{{C}{D}}}}
 \begin{bmatrix}
  {\acute{{w}}}_1\\
  \vdots\\
  {\acute{{w}}}_{{\ell}}
 \end{bmatrix},\end{aligned}$$ using the reproducing property. This derivation shows that the operator $B = {{\ensuremath{K}}_{{C}{D}}}\in{\mathbb{R}}^{{M}\times{\ell}}$, the kernel matrix between the data ${C}$ generating the atoms ${\mathcal{F}_{{C}}}$ of ${\ensuremath{{\mathcal{H}}_{{C}}}}$ and the data ${D}$ generating the atoms ${\mathcal{F}_{{D}}}$ of ${\ensuremath{{\mathcal{H}}_{{D}}}}$. Using similar arguments, it can be shown that, given sensing locations ${X}= \{x_1,x_2,\dots, x_{{N}}\}$, ${{\ensuremath{K}}_{{D}}}\in{\mathbb{R}}^{{N}\times{\ell}}$ is the kernel matrix between ${X}$ and ${D}$. Thus the finite-dimensional evolution equations equivalent to [(\[eq:ideal\_lin\_evol\_control\])]{} are $$\begin{aligned}
 {w}_{k} &= A{w}_k + {{\ensuremath{K}}_{{C}{D}}}{\acute{{w}}}_k{\label{eq:k_measure_c}}\\
 y_{k} &= {\ensuremath{K}}_{k} w_{k} {\label{eq:k_measure_c1}}.\end{aligned}$$ We pause here to point out just how flexible the kernel-based framework is. First of all, the choice of kernel completely determines the space ${\mathcal{H}}$, which may allow wildly different functional outputs for 