using polynomials as lower / upper bound?

Question

I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials $p_1(\vec{x})$ and $p_2(\vec{x})$ such that

$p_1(\vec{x}) \leq f(\vec{x}) \leq p_2(\vec{x}) ~~ \forall \vec{x} \in D $.

Ideally the polynomials $p_1$ and $p_2$ are defined as a linear combination of a few polynomials from some polynomial basis, and the bounds can be made arbitrarily tight by including more basis elements.

Any well studied objects of this sort would be ideal, I've looked at some basic polynomial approximation techniques but they only provide approximations (which can be re-purposed for bounds), but if there are any literature concerning just the bounds it would be a good starting point.

Answer 1

Let me turn my comments into an answer. I originally thought that the question would be off-topic here, so I chose not to give a full answer.

Let $f:D\to\mathbb R$ be any function on a bounded domain $D$. If there are polynomials $p_1,p_2$ so that $p_1\leq f\leq p_2$ on $D$, then $f$ is bounded. On the other hand, if $f$ is bounded, you can use constant polynomials for $p_1$ and $p_2$. Thus such bounding polynomials exist if and only if $f$ is bounded.

Estimating a continuous function on $\bar D$ arbitrarily tightly by polynomials is possible. (See this proof, for example.) This problem is actually equivalent with estimating function uniformly with polynomials: Tight polynomial upper and lower bounds provide a good polynomial approximation of $f$. In the other direction, if you estimate the functions $f\pm\epsilon$ with error $\epsilon$, you get polynomials above and below $f$ with distance at most $2\epsilon$ to it. (You can actually choose $p_2=p_1+2\epsilon$ if you do it like this.)

I am not an expert on polynomial approximation, so I can't recommend any literature. But there is certainly literature on polynomial approximation, and with the above argument you can turn polynomial approximation into polynomial bounds.