2010/03/17 by J. M. Almira, Almira, J. M., Timur Oikhberg +2 · 2 citations
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #math.CA
paper · pdf · doi:10.48550/arxiv.1003.3411
41 pages, Submitted to a Journal. A natural continuation of this paper is also downloadable at Arxiv: See J. M. Almira and T. Oikhberg, "Shapiro's theorem for Subspaces", at arXiv:1009.5535v1
openalex publication_date 2010/03/17 · arxiv created 2010/10/25 · arxiv updated 2010/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An approximation scheme is a family of homogeneous subsets (An) of a quasi-Banach space X, such that A1 \subsetneq A2 \subsetneq ... \subsetneq X, An + An ⊂ AK(n), and ∪n An = X. Continuing the line of research originating at a classical paper by S.N. Bernstein (in 1938), we give several characterizations of the approximation schemes with the property that, for every sequence \εn\\searrow 0, there exists x∈ X such that dist(x,An)≠ O(εn) (in this case we say that (X,\An\) satisfies Shapiro's Theorem). If X is a Banach space, x ∈ X as above exists if and only if, for every sequence \δn\ \searrow 0, there exists y ∈ X such that dist(y,An) ≥ δn. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.