A negative result on multivariate convex approximation by positive linear operators (Q1919087)

The author considers the problem of multivariate polynomial convex approximation by linear positive operators. One denotes by \(E\) a \(k\)-dimensional compact convex set in \(\mathbb{R}^k\) with \(k\geq2\) and \(\Omega\) an open set containing \(E\). Let \(L_n: C(E)\to C^1(\Omega)\) be a sequence of linear positive operators. The main result of this paper consists in showing that if \(L_n\) preserves convexity and is invariant on affine functions, i.e. \(L_n\ell=\ell\) for all \(\ell\) in the space \(\mathbb{P}_1\) of affine functions, then \(L_n\) is trivial (that is \(L_n f\in \mathbb{P}_1\) on \(E\) for all \(f\in C(E)\)) and \(E\) is a simplex. This result represents an essential difference between the cases of univariate and multivariate positive linear polynomial operators in the case of shape preserving approximation. Such a difference was first remarked by \textit{G. Z. Chang} and \textit{P. J. Davis} [J. Approximation Theory 40, 11-28 (1984; Zbl 0528.41005)] in the case of Bernstein polynomial operators.