Reproducing kernel Hilbert space compactification of unitary evolution groups

A framework for coherent pattern extraction and prediction of observables measure-preserving, ergodic dynamical systems with both atomic continuous spectral components is developed. This based on an approximation the generator system by a compact operator Wτ reproducing kernel Hilbert space (RKHS). The skew-adjoint, thus can be represented projection-valued measure, discrete compactness, associated orthonormal basis eigenfunctions. These eigenfunctions are ordered in terms Dirichlet energy, provide notion under dynamics akin to Koopman part spectrum. In addition, generates unitary evolution group {etWτ}t∈R RKHS, which approximates system. We establish convergence results spectrum Borel functional calculus as τ→0+, well data-driven formulation utilizing time series data. Numerical applications spectra, namely torus rotation, Lorenz 63 system, Rössler presented.