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Given $k\in N$, a nonnegative function $f\in C^r[a,b]$, $r\ge 0$, anarbitrary finite collection of points  $\big\{\alpha_i\big\}_{i\in J} \subset [a,b]$, and a corresponding collectionof nonnegative integers  $\big\{m_i\big\}_{i\in J}$ 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) $|f(x)-P_n(x)| \le c \rho_n^r(x) \omega_k(f^{(r)}, \rho_n(x); [a,b])$,$x\in [a,b]$, where $\rho_n (x):= n^{-1} \sqrt{1-x^2} +n^{-2}$ and $\omega_k$is the classical $k$-th modulus of smoothness,  (ii) $P^{(\nu)}(\alpha_i)=f^{(\nu)}(\alpha_i)$, for all $0\le \nu \le m_i$and all $i\in J$,  and  (iii) either $P \ge f$ on $[a,b]$ (\emph{onesided} approximation), or $P \ge0$ on $[a,b]$ (\emph{positive} approximation)?  We provide {\em precise answers} not only to this question, but also tosimilar questions for more general {\em intertwining} and {\em copositive}polynomial approximation. It turns out that many of these answers are quiteunexpected.  We also show that, in general, similar questions for $q$-monotoneapproximation with $q\ge 1$ have negative answers, i.e., $q$-monotoneapproximation with general interpolatory constraints is impossible if $q\ge 1$.

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