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We establish the convergence of a class of numerical algorithms, known asDynamic Mode Decomposition (DMD), for computation of the eigenvalues andeigenfunctions of the infinite-dimensional Koopman operator. The algorithms acton data coming from observables on a state space, arranged in Hankel-typematrices. The proofs utilize the assumption that the underlying dynamicalsystem is ergodic. This includes the classical measure-preserving systems, aswell as systems whose attractors support a physical measure. Our approachrelies on the observation that vector projections in DMD can be used toapproximate the function projections by the virtue of Birkhoff's ergodictheorem. Using this fact, we show that applying DMD to Hankel data matrices inthe limit of infinite-time observations yields the true Koopman eigenfunctionsand eigenvalues. We also show that the Singular Value Decomposition, which isthe central part of most DMD algorithms, converges to the Proper OrthogonalDecomposition of observables. We use this result to obtain a representation ofthe dynamics of systems with continuous spectrum based on the lifting of thecoordinates to the space of observables. The numerical application of thesemethods is demonstrated using well-known dynamical systems and examples fromcomputational fluid dynamics.

Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator