  A methodological framework for ensemble-based estimation and simulation of
high dimensional dynamical systems such as the oceanic or atmospheric flows is
proposed. To that end, the dynamical system is embedded in a family of
reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the
dynamics. In the RKHS family, the Koopman and Perron-Frobenius operators are
unitary and uniformly continuous. This property warrants they can be expressed
in exponential series of diagonalizable bounded evolution operators defined
from their infinitesimal generators. Access to Lyapunov exponents and to exact
ensemble based expressions of the tangent linear dynamics are directly
available as well. The RKHS family enables us the devise of strikingly simple
ensemble data assimilation methods for trajectory reconstructions in terms of
constant-in-time linear combinations of trajectory samples. Such an
embarrassingly simple strategy is made possible through a fully justified
superposition principle ensuing from several fundamental theorems.
