Mean Dimension of Bernstein Spaces and Universal Real Flows

The paper proves universality of Y_{α,β} (Theorem 1.5) via a three-step harmonic-analytic embedding: (i) embed X into C(R)^N by the orbit map, (ii) convolve with a time-domain partition of unity {ξ_n} whose sum is 1 to get band-limited coordinates and injectivity (Lemma 5.2), and (iii) add a Lipschitz regularization step to land in B(·)∩L(R) while preserving injectivity; this completes Theorem 1.5 (Section 5) . The candidate solution gives a different, self-contained construction: it builds a frequency-space C^∞ partition of unity {w_{m,n}}, convolves orbit functions with Schwartz kernels, scales to enforce 1-Lipschitz bounds, and proves injectivity by summing frequency multipliers over infinitely many m with p(m)=k to force F(h)=0 on all of R. Both approaches are rigorous and culminate in an equivariant topological embedding into Y_{α,β}.