ideas from systems theory for knowing necessary conditions for functional convergence; furthermore, it offers a natural framework for designing control mechanisms as well.

![Two types of Hilbert space evolutions. Left: the model, represented by the functions $m_i$, switches discretely in the Hilbert space ${\mathcal{H}}$; Right: the evolution of the function $m_t$ is smooth, represented by a solution to an ordinary differential equation in ${\mathcal{H}}$.[]{data-label="fig:hilbert_evolution"}](model.png){width="0.8\columnwidth"}

In this paper, we restrict our attention to the class of functional evolutions $\mathbb{F}$ defined by linear Markovian transitions in an RKHS. While extension to the nonlinear case is possible (and non-trivial), it is not pursued in this paper to help ease the exposition of key ideas. Let $y\in{\mathbb{R}}^{{N}}$ be the measurements of the function available from ${N}$ sensors, ${\mathcal{A}}:{\mathcal{H}}\to{\mathcal{H}}$ be a linear transition operator in the RKHS ${\mathcal{H}}$, and ${\mathcal{K}}:{\mathcal{H}}\to{\mathbb{R}}^{{N}}$ be a linear measurement operator, the model for the infinite-dimensional functional evolution and measurement studied in this paper is: $$\begin{aligned}
{\label{eq:ideal_lin_evol}}
 f_{k+1} &= {\mathcal{A}}f_k + \eta_k\\
 y_k &= {\mathcal{K}}f_k + \zeta_k,\end{aligned}$$ where $\eta_k$ is a zero-mean stochastic process in ${\mathcal{H}}$, and $\zeta_k$ is a Wiener process in ${\mathbb{R}}^{{N}}$. For many kernels, the feature map ${\psi}$ is unknown, and therefore it is necessary to work in 