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Mathematics (CMC)

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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 = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = S∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there exists x ∈ X and no ∈ N such that

en/3 ≤ dist(x, Vn) ≤ 3en

for any n ≥ no. 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.


Final published article in the Journal of Approximation Theory.

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© 2016 Elsevier Inc.

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