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 $\{{\alpha }_i{\}}_{i\in J}\subset [a,b]$, and a corresponding collection of nonnegative integers $\{m_i{\}}_{i\in J}$ 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 $q\ge 1$.