2403.02438
Approximation of the Koopman operator via Bernstein polynomials
Rishikesh Yadav, Alexandre Mauroy
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the multivariate bound ‖BnKf − Kf‖∞ ≤ (3/2) Ωf(Lφ √(∑l 1/nl)) restricted to φ([0,1]m) (Theorem 4.3), via a Bernstein–modulus-of-continuity argument and a Lipschitz pullback of the modulus (Lemma 2.2). The candidate solution derives exactly the same inequality using the probabilistic representation of multivariate Bernstein polynomials, the same 3/2 optimization by choosing δ = √(∑l 1/nl), and the same Lipschitz transfer Ωg(δ) ≤ Ωf(Lφ δ)|φ([0,1]m). The logic, hypotheses, and constant match the paper’s result and proof strategy closely (Theorem 4.3 and its proof; Lemma 2.2).
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript rigorously adapts classical Bernstein approximation tools to approximate the Koopman operator and derives explicit uniform error bounds, including the multivariate result central to this audit. The development is correct and well-motivated; numerical illustrations are informative. Minor clarifications about norm conventions, modulus saturation, and the role of the probabilistic representation would further improve readability and reproducibility.