A Negative Result on Multivariate Convex Approximation by Positive Linear Operators

✍ Scribed by Xuguang Lu

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 516 KB
• Volume: 86
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.0057

✦ Synopsis

We consider the problem of multivariate convex approximation by positive linear operators. Let E be a k-dimensional compact convex set in R k with k 2, 0/R k an open set containing E, and let L: C(E ) Ä C 1 (0) be a positive linear operator. Our main result of this paper shows that if L preserves convexity and satisfies Ll=l on E for all l # P 1 (the space of affine functions), then L is trivial (i.e., Lf # P 1 on E for all f # C(E)) and E is a simplex.