$L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition

Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by using the kernel's canonical features and, thus, data-informed observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.