Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds

The paper proves an existence result for the joint limit (Theorem 5.1) by combining almost-sure data-limit convergence at fixed N (Theorem 3.5/Corollary 3.6) with dictionary-limit convergence (Theorem 4.2) and then, under additional boundedness assumptions, supplies high-probability finite-sample error bounds (Proposition 5.6; Theorem 5.7). The candidate solution also establishes the joint-limit convergence, but via a different route: an explicit deterministic schedule M_N based on union-of-Chebyshev bounds and Borel–Cantelli, plus a perturbation analysis of Ĝ and Ĉ leading to operator-norm control of Â_{N,M}−A_N, and then the same projection argument to reach A. This is consistent with the paper’s setting and assumptions (Assumptions 1–4) and yields a slightly stronger existence statement (deterministic schedule) than Theorem 5.1, though it does not deliver the paper’s quantitative rates. Minor omissions in the model (e.g., not explicitly stating the continuity-evaluability condition of Assumption 1(b)) do not affect correctness of the main claim. Overall, both arguments are correct and structurally different; the paper also provides sharper probabilistic rates that the model does not.