Even if the solution to this problem is not known, I would be happy to be directed to a subfield which deals in problems of this type. For the sake of specificity, I will give an instance of the problem I am interested in.
Given a set $S$ of $n$ points from $[-1, 1]\times\{0, 1\}$, such that each tuple in $S$ has a distinct first element, there exists a real polynomial interpolating between these points whose degree is of order $n$; this is well known.
We want to impose the additional constraint that, on the interval $[-1, 1]$, the modulus of a (possibly complex) polynomial $f$ interpolating between the points in the set specified above does not itself exceed 1. Clearly in some cases the same polynomial will suffice, but not in all cases.
My questions are these; (1) how does one prove that such a polynomial $f$ does exist, of any degree, and (2), how does the degree of the required polynomial, if it exists (and I believe it does), depend on, for instance, a promised gap of at least $\delta$ between all first coordinates of the $n$ interpolation points in $S$.
If this minimum promised gap $\delta$ becomes very small, it is clear that an interpolating polynomial will require a very high derivative, which will tend to cause it to overshoot the modulus bound on the interval (and thus higher-degree corrections will be necessary).
While it is sometimes sufficient to use Hermite interpolation and specify derivatives of zero at interpolation points (constructing an $f$ which 'weaves' between these maxima and minima), this clearly does not capture the on-interval bounds in which I am interested.