  Koopman theory studies dynamical systems in terms of operator theoretic
properties of the Perron-Frobenius operator $\mathcal{P}$ and Koopman operator
$\mathcal{U}$ respectively. In this paper, we derive the rates of convergence
of approximations of $\mathcal{P}$ or $\mathcal{U}$ that are generated by
finite dimensional bases like wavelets, multiwavelets, and eigenfunctions, as
well as approaches that use samples of the input and output of the system in
conjunction with these bases. We introduce a general class of priors that
describe the information available for constructing such approximations and
facilitate the error estimates in many applications of interest. These priors
are defined in terms of the action of $\mathcal{P}$ or $\mathcal{U}$ on certain
linear approximation spaces. The rates of convergence for the estimates of
these operators are investigated under a variety of situations that are
motivated from associated assumptions in practical applications. When the
estimates of these operators are generated by samples, it is shown that the
error in approximation of the Perron-Frobenius or Koopman operators can be
decomposed into two parts, the approximation error and the sampling error. This
result emphasizes that sample-based estimates of Perron-Frobenius and Koopman
operators are subject to the well-known trade-off between the bias and variance
that contribute to the error, a balance that also features in nonlinear
regression and statistical learning theory.
