  We show that spectral data of the Koopman operator arising from an analytic
expanding circle map $\tau$ can be effectively calculated using an EDMD-type
algorithm combining a collocation method of order m with a Galerkin method of
order n. The main result is that if $m \geq \delta n$, where $\delta$ is an
explicitly given positive number quantifying by how much $\tau$ expands
concentric annuli containing the unit circle, then the method converges and
approximates the spectrum of the Koopman operator, taken to be acting on a
space of analytic hyperfunctions, exponentially fast in n. Additionally, these
results extend to more general expansive maps on suitable annuli containing the
unit circle.
