Analytic Extended Dynamic Mode Decomposition

We aim at developing an EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. Our method relies on an orthogonal projection on polynomial subspaces, which is equivalent to Taylor approximation in a data-driven setting. In the case of dynamics with a hyperbolic equilibrium, the method demonstrates excellent performance to capture the lattice structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation.