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Extended Dynamic Mode Decomposition (EDMD) is a data-driven tool for
forecasting and model reduction of dynamics, which has been extensively taken
up in the physical sciences. While the method is conceptually simple, in
deterministic chaos it is unclear what its properties are or even what it
converges to. In particular, it is not clear how EDMD's least-squares
approximation treats the classes of regular functions needed to make sense of
chaotic dynamics.
In this paper we develop a general, rigorous theory of EDMD on the simplest
examples of chaotic maps: analytic expanding maps of the circle. Proving a new
result in the theory of orthogonal polynomials on the unit circle (OPUC), we
show that in the infinite-data limit, the least-squares projection is
exponentially efficient for polynomial observable dictionaries. As a result, we
show that the forecasts and Koopman spectral data produced using EDMD in this
setting converge to the physically meaningful limits, at an exponential rate.
This demonstrates that with only a relatively small polynomial dictionary,
EDMD can be very effective, even when the sampling measure is not uniform.
Furthermore, our OPUC result suggests that data-based least-squares projections
may be a very effective approximation strategy.
