  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 differentiable functions on which chaotic
systems act.
  We develop for the first time a general, rigorous theory of EDMD on the
simplest examples of chaotic maps: analytic expanding maps of the circle. To do
this, we prove a new, basic approximation result in the theory of orthogonal
polynomials on the unit circle (OPUC) and apply methods from transfer operator
theory. We show that in the infinite-data limit, the least-squares projection
error is exponentially small for trigonometric polynomial observable
dictionaries. As a result, we show that forecasts and Koopman spectral data
produced using EDMD in this setting converge to the physically meaningful
limits, exponentially fast with respect to the size of the dictionary. 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 projection may be a very
effective approximation strategy more generally.
