  We rigorously derive novel error bounds for extended dynamic mode
decomposition (EDMD) to approximate the Koopman operator for discrete- and
continuous time (stochastic) systems; both for i.i.d. and ergodic sampling
under non-restrictive assumptions. We show exponential convergence rates for
i.i.d. sampling and provide the first superlinear convergence rates for ergodic
sampling of deterministic systems. The proofs are based on novel exact variance
representations for the empirical estimators of mass and stiffness matrix.
Moreover, we verify the accuracy of the derived error bounds and convergence
rates by means of numerical simulations for highly-complex dynamical systems
including a nonlinear partial differential equation.
