Universal properties of approximation operators

JOURNAL OF APPROXIMATION THEORY

卷:164

Article

Vogt, Andreas

关键词: Universal functions;    Approximation operators;    Korovkin-type approximation;   

DOI  :  10.1016/j.jat.2011.11.003

来源: Elsevier

【 摘 要 】

We discuss universal properties of some operators L-n : C[0, 1] -> C[0, 1]. The operators considered are closely related to a theorem of Korovkin (1960) [4] which states that a sequence of positive linear operators L-n on C[0, 1] is an approximation process if L(n)f(i) -> f(i) (n -> infinity) uniformly for i = 0, 1, 2, where f(i)(x) = x(i) We show that L(n)f may diverge in a maximal way if any requirement concerning L-n in this theorem is removed. There exists for example a continuous function f such that (L(n)f)(n is an element of N) is dense in (C[0, 1], parallel to.parallel to(infinity)), even if L-n is positive, linear and satisfies L-n P -> P (n ->infinity) for all polynomials P with P(0) = 0. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free