Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict positivity, in case it matters.

Consider all polynomials of $f$, such as $3f^4 - f^2 +2$. These are in $L^2(I)$, since $f$ is (essentially) bounded and $I$ is bounded. Let $P_f$ be the space of all such polynomials.

What can one say about the closure of $P_f$ in $L^2(I)$, that is $\overline{P_f}^{L^2(I)}$?

Extreme examples

1. If $f$ is constant (on a given subset of $I$), then so are all of its polynomials.
2. If $f(x) \equiv x$, then $P_f$ is the set of all polynomials, which is dense in $L^2(I)$.