A priori bound on Koopman eigenfunctions for applying Proposition 1

Determine an explicit a priori bound M on the supremum norm of an analytic Koopman eigenfunction φ_λ over the domain D^n(S), i.e., establish M such that sup_{x ∈ D^n(S)} |φ_λ(x)| ≤ M, so that the error estimate in Proposition 1 for |φ_λ(x) − (P_N φ_λ)(x)| can be applied without requiring additional coefficient-based assumptions.

Background

Proposition 1 provides an error bound for approximating an analytic function (here, a Koopman eigenfunction φ_λ) by its truncated Taylor series, provided an a priori uniform bound M on the function over the domain D^n(S) is available.

The authors note that this required a priori bound on φ_λ is not available in general. They therefore propose alternative bounds that rely on estimates of the Taylor coefficients (Propositions 2 and 3), but the direct a priori supremum bound remains an explicit unknown.