Reproducing kernel Hilbert space compactification of unitary evolution groups

• Published in: Applied and computational harmonic analysis Vol. 54; pp. 75 - 136
• Main Authors: Das, Suddhasattwa, Giannakis, Dimitrios, Slawinska, Joanna
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2021
• Subjects: Ergodic dynamical systems; Koopman operators; Perron-Frobenius operators; Reproducing kernel Hilbert spaces; Spectral theory; Perron-Frobenius operators; Ergodic dynamical systems; Koopman operators; Reproducing kernel Hilbert spaces; Spectral theory
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2021.02.004

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