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 $\|\nu-1\|_{\mathcal F}$ is exponentially small in $1/\varepsilon$, then every $0\leq g\leq\nu$ has a dense model $0\leq\widetilde g\leq1$ satisfying
$$
\|g-\widetilde g\|_{\mathcal F}\leq\varepsilon.
$$
The 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.

Dense model theorem for a multiplicative test family

Use the normalized <inner product>
$$
\langle g,h\rangle=\mathbb E_{n\leq N}g(n)h(n)=\frac1N\sum_{n\leq N}g(n)h(n).
$$
The <test-function seminorm> generated by $\mathcal F$ and its <dual test-function norm> are
$$
\|g\|_{\mathcal F}=\sup_{f\in\mathcal F}|\langle g,f\rangle|,
\qquad
\|g\|_{\mathcal F}^{*}=\sup_{\|h\|_{\mathcal F}\leq1}|\langle g,h\rangle|.
$$
The first quantity may only be a <seminorm> if $\mathcal F$ does not separate all functions in $\mathcal H_N$; correspondingly, the second may be infinite outside the linear span detected by $\mathcal F$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 4 / b / Solution

{title2=$\|\cdot\|_{\mathcal F}^{*}$}

The dual test-function norm is
$$
\|g\|_{\mathcal F}^{*}=\sup_{\|h\|_{\mathcal F}\leq1}|\langle g,h\rangle|.
$$
If $\mathcal F$ is closed, convex, and symmetric, the <Bipolar theorem for a dual pair> identifies its dual unit ball with $\mathcal F$. Submultiplicativity under pointwise products allows a polynomial in one test function to remain controlled in this norm.

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