Finite-dimensional approximations of push-forwards on locally analytic functionals

The paper proves the Frobenius-norm error bound for the truncated jet push-forward estimator via a Fock-space/Exp-space argument combined with spectral bounds for Hankel-type moment matrices, yielding ||C_m − Ĉ_{p,m,n,ZN}||_Fr ≲ (√m!/ε) σ_{p−p′,r}^n n^{d/8} |||·−p|_{K0}^n||_{L^2(μ̂N)}. The candidate solution derives the same rate and structure by a different route: an explicit coefficient-tail decomposition through a two-variable generating function (Faà di Bruno/Cauchy) together with a Nikolskii/Christoffel-type bound for the basis tail, and a stability step using the empirical/population Gram closeness. The end result matches the paper’s theorem under the same assumptions, though the model compresses several technical lemmas (e.g., precise coefficient tail and tail-of-basis estimates) that the paper establishes via Lemma 3.8, Lemma 4.9, and Appendix A. Hence, both are correct, with different proof styles.