A NEGATIVE RESULT ON MULTIVARIATE CONVEX APPROXIMATION BY POSITIVE LINEAR-OPERATORS

We consider the problem of multivariate convex approximation by positi ve linear operators. Let E be a k-dimensional compact convex set in R( k) with k greater than or equal to 2, Omega subset of R(k) an open set containing E, and let L: C(E) --> C-1(Omega) be a positive linear ope rator. Our main result of this paper shows that if L preserves convexi ty and satisfies Ll = l on E for all l is an element of P-1 (the space of affine functions), then L. is trivial (i.e., Lf is an element of P -1 on E For all f is an element of C(E)) and E is a simplex. (C) 1996 Academic Press, Inc.