  A framework for coherent pattern extraction and prediction of observables of
measure-preserving, ergodic dynamical systems with both atomic and continuous
spectral components is developed. It is based on an approximation of the
generator of the system by a compact operator $W_\tau$ on a reproducing kernel
Hilbert space (RKHS). A key tool is that $W_\tau$ is skew-adjoint , and thus
can be characterized by a unique projection-valued measure, discrete by
compactness, and an associated orthonormal basis of eigenfunctions. These
eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and
provide a notion of coherent observables under the dynamics akin to the Koopman
eigenfunctions associated with the atomic part of the spectrum. In addition,
the regularized generator has a well-defined Borel functional calculus allowing
the construction of a unitary evolution group $\{ e^{t W_\tau}
\}_{t\in\mathbb{R}}$ on the RKHS, which approximates the unitary Koopman
evolution group of the original system. We establish convergence results for
the spectrum and Borel functional calculus of the regularized generator to
those of the original system in the limit $\tau \to 0^+$. Convergence results
are also established for a data-driven formulation, where these operators are
approximated using finite-rank operators obtained from observed time series. An
advantage of working with an RKHS structure is that one can perform pointwise
evaluation and interpolation through bounded linear operators, which is not
possible in $L^p$ spaces. This enables the out of sample evaluation of
data-approximated eigenfunctions, as well as data-driven forecasts initialized
with pointwise initial data (as opposed to probability densities in $L^p$). The
pattern extraction and prediction framework is numerically applied to three
instances of ergodic dynamical systems with atomic and continuous spectra.
