Dense model theorem for a multiplicative test family

Let $\mathcal{F}$ be a closed, convex, symmetric family of functions into $[-1,1]$ that contains the constant function $1$, and suppose its dual test-function norm is submultiplicative under pointwise products. If a nonnegative majorant $\nu$ has average at most one and $\Vert \nu -1{\Vert }_{\mathcal{F}}$ is exponentially small in $1/\epsilon$, then every $0\le g\le \nu$ has a dense model $0\le \tilde{g}\le 1$ satisfyingThe proof separates $g$ from the convex set of dense models, then approximates the positive part of the separating test by a polynomial. Submultiplicativity controls every power in that polynomial.

On any fixed compact interval, the positive part of a real-valued function $t_{+}=\max (0,t)$ can be approximated uniformly by a real polynomial. Quantitative dense-model arguments use a version whose degree and coefficient growth are explicitly controlled in terms of the approximation error.