  We establish the convergence of a class of numerical algorithms, known as
Dynamic Mode Decomposition (DMD), for computation of the eigenvalues and
eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act
on data coming from observables on a state space, arranged in Hankel-type
matrices. The proofs utilize the assumption that the underlying dynamical
system is ergodic. This includes the classical measure-preserving systems, as
well as systems whose attractors support a physical measure. Our approach
relies on the observation that vector projections in DMD can be used to
approximate the function projections by the virtue of Birkhoff's ergodic
theorem. Using this fact, we show that applying DMD to Hankel data matrices in
the limit of infinite-time observations yields the true Koopman eigenfunctions
and eigenvalues. We also show that the Singular Value Decomposition, which is
the central part of most DMD algorithms, converges to the Proper Orthogonal
Decomposition of observables. We use this result to obtain a representation of
the dynamics of systems with continuous spectrum based on the lifting of the
coordinates to the space of observables. The numerical application of these
methods is demonstrated using well-known dynamical systems and examples from
computational fluid dynamics.
