Koopman spectra in reproducing kernel Hilbert spaces

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the L² space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on N states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as N→∞. In that case, the corresponding Fourier frequency is also a Koopman eigenfrequency, modulo a unique translate by a Nyquist frequency interval, and the RKHS extensions of the N-sample Fourier functions converge to a Koopman eigenfunction in RKHS norm. For Koopman eigenfunctions in L² that do not have RKHS representatives, the RKHS extensions of Fourier functions at the corresponding eigenfrequencies are shown to converge in L² norm. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.