It has recently been observed that eigenvalues and modes of the Koopman operator may be approximated using a data-driven algorithm called Dynamic Mode Decomposition (DMD). We first provide a new definition of DMD that is equivalent to the original, but is more amenable to analysis. We provide a theorem that characterizes when Koopman eigenvalues appear as DMD eigenvalues, and elucidates some limitations of DMD when applied to nonlinear systems. We then describe an Extended DMD approach that approximates the Koopman operator directly, using a user-specified set of basis functions. We also show how Extended DMD can be rewritten as a kernel method, which enables Koopman-based computation in high-dimensional systems. We illustrate with a number of examples, including an application using the Koopman eigenfunctions as a set of "intrinsic coordinates" that enable a data-fusion/state-reconstruction task to be accomplished.
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