Bernstein’s Lethargy Theorem in Fréchet Spaces

Authors

Asuman Güven Aksoy, Claremont McKenna CollegeFollow
Grzegorz Lewicki, Jagiellonian University, Poland

Document Type

Article

Department

Mathematics (CMC)

Publication Date

2016

Abstract

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {V_n} be a nested sequence of subspaces of X such that V_n ⊆ V_n+1 for any n ∈ N and X = S∞ _n=1 V_n. Let e_n be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, V_n)}, we prove that there exists x ∈ X and no ∈ N such that

e_n/3 ≤ dist(x, V_n) ≤ 3e_n

for any n ≥ n_o. By using the above theorem, we prove both Shapiro’s and Tyuremskikh’s theorems for Fréchet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fréchet spaces will be discussed. We also give a theorem improving Konyagin’s result for Banach spaces.

Comments

Final published article in the Journal of Approximation Theory.

Rights Information

© 2016 Elsevier Inc.

DOI

10.1016/j.jat.2016.05.003

Recommended Citation

A. G. Aksoy, G. Lewicki, Bernstein lethargy theorem for Frechet spaces, Journal of Approximation Theory, 209, pp. 58{77, 2016, DOI: 10.1016/j.jat.2016.05.003.