Bernstein's Lethargy Theorem in Frechet spaces

JOURNAL OF APPROXIMATION THEORY

卷:209

Article

Aksoy, Asuman Guven¹  Lewicki, Grzegorz² 

[1] Claremont Mckenna Coll, Dept Math, Claremont, CA 91711 USA

[2] Jagiellonian Univ, Dept Math, Lojasiewicza 6, PL-30348 Krakow, Poland

关键词: Best approximation;    Bernstein's Lethargy Theorem;    Frechet spaces;   

DOI  :  10.1016/j.jat.2016.05.003

来源: Elsevier

【 摘 要 】

In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Frechet spaces. Let X be an infinite-dimensional Frechet space and let V = {V-n} be a nested sequence of subspaces of X such that V-n subset of Vn+1 for any n is an element of N. Let en be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup{dist(x, V-n)}, we prove that there exist x is an element of X and n(o) is an element of N such that en/3 <= dist (x, V-n) <= 3e(n) for any n >= n(o). By using the above theorem, as a corollary we obtain classical Shapiro's (1964) and Tyuriemskih's (1967) theorems for Banach spaces. Also we prove versions of both Shapiro's (1964) and Tyuriemskih's (1967) theorems for Frechet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Frechet spaces will be discussed. We also give a theorem improving Konya gin's (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Frechet spaces, such as Orlicz spaces generated by s-convex functions and locally bounded Frechet spaces. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free