  The Koopman operator approach provides a powerful linear description of
nonlinear dynamical systems in terms of the evolution of observables. While the
operator is typically infinite-dimensional, it is crucial to develop
finite-dimensional approximation methods and characterize the related
approximation errors with upper bounds, preferably expressed in the uniform
norm. In this paper, we depart from the traditional use of orthogonal
projection or truncation, and propose a novel method based on Bernstein
polynomial approximation. Considering a basis of Bernstein polynomials, we
construct a matrix approximation of the Koopman operator in a computationally
effective way. Building on results of approximation theory, we characterize the
rates of convergence and the upper bounds of the error in various contexts
including the cases of univariate and multivariate systems, and continuous and
differentiable observables. The obtained bounds are expressed in the uniform
norm in terms of the modulus of continuity of the observables. Finally, the
method is extended to a data-driven setting through a proper change of
coordinates, where it is shown to demonstrate good performance for trajectory
prediction.
