Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints

Given $k\in N$, a nonnegative function $f\in C^r[a,b]$, $r\ge 0$, an arbitrary finite collection of points ${\left\{{\alpha }_i\right\}}_{i\in J}\subset [a,b]$, and a corresponding collection of nonnegative integers ${\left\{m_i\right\}}_{i\in J}$ with $0\le m_i\le r$, $i\in J$, is it true that, for sufficiently large $n\in N$, there exists a polynomial $P_n$ of degree $n$ such that

(i) $|f\left(x\right)-P_n\left(x\right)|\le c{\rho }_n^r\left(x\right){\omega }_k(f^{\left(r\right)},{\rho }_n\left(x\right);[a,b])$, $x\in [a,b]$, where ${\rho }_n\left(x\right)≔n^{-1}\sqrt{1-x^2}+n^{-2}$ and ${\omega }_k$ is the classical $k$th modulus of smoothness.

(ii) $P^{\left(\nu \right)}\left({\alpha }_i\right)=f^{\left(\nu \right)}\left({\alpha }_i\right)$, for all $0\le \nu \le m_i$ and all $i\in J$,

and

(iii) either $P\ge f$ on $[a,b]$ (onesided approximation), or $P\ge 0$ on $[a,b]$ (positive approximation)?

We provide precise answers not only to this question, but also to similar questions for more general intertwining and copositive polynomial approximation. It turns out that many of these answers are quite unexpected.

We also show that, in general, similar questions for $q$-monotone approximation with $q\ge 1$ have negative answers, i.e., $q$-monotone approximation with general interpolatory constraints is impossible if $q\ge 1$.