We discuss a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman group of unitary operators governing dynamical evolution in a smooth orthonormal basis acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection-diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also use a special property of this basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for arbitrary probability densities and observables. We present applications to dynamical systems on tori and satellite observations of atmospheric convection.