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Extended Dynamic Mode Decomposition (EDMD) is a data-driven tool forforecasting and model reduction of dynamics, which has been extensively takenup in the physical sciences. While the method is conceptually simple, indeterministic chaos it is unclear what its properties are or even what itconverges to. In particular, it is not clear how EDMD's least-squaresapproximation treats the classes of differentiable functions on which chaoticsystems act.  We develop for the first time a general, rigorous theory of EDMD on thesimplest examples of chaotic maps: analytic expanding maps of the circle. To dothis, we prove a new, basic approximation result in the theory of orthogonalpolynomials on the unit circle (OPUC) and apply methods from transfer operatortheory. We show that in the infinite-data limit, the least-squares projectionerror is exponentially small for trigonometric polynomial observabledictionaries. As a result, we show that forecasts and Koopman spectral dataproduced using EDMD in this setting converge to the physically meaningfullimits, exponentially fast with respect to the size of the dictionary. Thisdemonstrates that with only a relatively small polynomial dictionary, EDMD canbe very effective, even when the sampling measure is not uniform. Furthermore,our OPUC result suggests that data-based least-squares projection may be a veryeffective approximation strategy more generally.

Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos