Analytic Extended Dynamic Mode Decomposition

The paper establishes, using RKHS projection identities and Bauer–Fike perturbation bounds, that the data-driven Koopman matrix K̂ = X^T G^{-1} Y built from samples converges entrywise to the exact Taylor/Galerkin matrix, that eigenvalues of each degree block Krr converge to the lattice eigenvalues µ^α, and that the principal eigenfunction approximants converge strongly in H as M→∞ then N→∞ (Lemma 6.1, Proposition 6.2, Proposition 6.4) . The candidate solution proves the same two targets via: (i) interpreting f^T G^{-1} g as ⟨P_M f, P_M g⟩_H and showing P_M→I strongly under i.i.d. uniform sampling; (ii) resolvent/Riesz-projection continuity for spectral convergence of the diagonal blocks; and (iii) a recursive construction of principal-eigenfunction coefficients, with an explicit nonresonance assumption for invertibility of K̃_rr−µ_j I. The arguments line up with the paper’s structure (X^T G^{-1} Y, block-lower-triangularity, and the same recursion (3.7)) but take different technical routes (Riesz projections vs. Bauer–Fike) and the model is more explicit about the nonresonance needed to invert K_rr−µ̂_j I . The only substantive gap in the paper is that the inversion step for eigenfunction coefficients is used without stating a nonresonance condition; the model states and addresses this explicitly. Otherwise, both reach the same conclusions.