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

Abstract

Given $k\in N$, a nonnegative function $f\in C^r[a,b]$, $r\ge 0$, an arbitrary finite collection of points $\big\{\alpha_i\big\}_{i\in J} \subset [a,b]$, and a corresponding collection of 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 $P \ge 0$ on $[a,b]$ (\emph{positive} approximation)? We provide {\em precise answers} not only to this question, but also to similar questions for more general {\em intertwining} and {\em 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 $q\ge 1$.