x_{{N}}\}}$, where $x_i\in{\Omega}\subset{\mathbb{R}}^d$, and $x_i\neq x_j$.

We can reuse Propositions \[prop:1\], \[prop:2\], and \[prop:3\] to prove kernel controllability results, because the structure of the control matrix ${{\ensuremath{K}}_{{C}{D}}}$ in [(\[eq:k\_measure\_c\])]{} is also that of a kernel matrix.

Kernel ${k}$, basis points ${\mathcal{C}}$, final time step $T_f$. $1)$ Sample data $\{y^i_k\}_{i=1}^{{N}}$ from $f(x,k)$. $2)$ Estimate ${\widehat{w}}_k$ via standard kernel inference procedure. $3)$ Store weights ${\widehat{w}}_k$ in matrix ${\mathcal{W}}\in{\mathbb{R}}^{{M}\times T_f}$. Infer ${\widehat{A}}$ using method of choice (e.g. matrix least squares). Compute the covariance matrix ${\widehat{B}}$ of the observed weights ${\mathcal{W}}$. estimated transition matrix ${\widehat{A}}$, predictive covariance matrix ${\widehat{B}}$.

Kernel ${k}$, basis points ${\mathcal{C}}$, estimated system matrix ${\widehat{A}}$, estimated covariance matrix ${\widehat{B}}$. Compute the cyclic index ${l}$ of ${\widehat{A}}$, and compute [(\[eq:empKShadFull\])]{}, by possibly iterating over ${\mathcal{X}}= {\{x_1, \dots, x_{{N}}\}}$. Use ${\widehat{A}}$, ${\widehat{B}}$, and ${\mathbf{{\ensuremath{K}}}}$ to initialize a state-observer (e.g. Kalman filter (KF)) on ${\ensuremath{{\mathcal{H}}_{{C}}}}$. 1) Sample data $\{y^i_k\}_{i=1}^{{N}}$ from $f(x,k)$. 2) Propagate KF estimate ${\widehat{w}}_{k+1}$ forward to time $t_f$, correct using measurement feedback with $\{y^i_k\}_{i=1}^{{N}}$. 3) Output predicted function $\widehat{f}(x,k+1)$ and predictive covariance of KF.

Kernel ${k}$, basis points ${\mathcal{C}}$, estimated system matrix ${\widehat{A}}$, estimated covariance matrix ${\widehat{B}}$, and function $f_{\text{ref}}$ to drive initial function to. (see Algorithm \[alg:egp\_inf\]). Use Jordan decomposition of 