  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.
