Universal polynomial majorants on convex bodies

JOURNAL OF APPROXIMATION THEORY

卷:111

Article

Kroó, A

关键词: convex bodies;    polynomial majorants;    polytopes;    polytopal approximation;   

DOI  :  10.1006/jath.2001.3573

来源: Elsevier

【 摘 要 】

Let K be a convex body in R-d (d greater than or equal to 2), and denote by B-n(K) the set of all polynomials p, in R-d of total degree less than or equal ton such that \p(n)\ less than or equal to 1 on K. In this paper we consider the following question: does there exist a p(n)* is an element of B-n(K) which majorates every clement of B-n(K) outside of K? In other words can we find a minimal gamma greater than or equal to 1 and p(n)* is an element of B-n(K) so that \p(n)(x)\ less than or equal to gamma \p(n)*(x)\ for every p(n) is an element of B-n(K) and x is an element of R-d\K? We discuss the magnitude of gamma and construct the universal majorants p(n)* for even n. It is shown that gamma can be 1 only on ellipsoids. Moreover, gamma =O(1) on polytopes and has at most polynomial growth with respect to it, in general, for every convex body K. (C) 2001 Academic Press.

【 授权许可】

Free