1. Approximation schemes satisfying Shapiro’s Theorem
- Author
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Timur Oikhberg and J. M. Almira
- Subjects
Mathematics(all) ,Numerical Analysis ,Sequence ,Approximation scheme ,Applied Mathematics ,General Mathematics ,Existential quantification ,Banach space ,Approximation by dictionary ,Space (mathematics) ,Combinatorics ,Approximation error ,Homogeneous ,Scheme (mathematics) ,Line (geometry) ,Bernstein’s Lethargy ,Analysis ,Mathematics - Abstract
An approximation scheme is a family of homogeneous subsets (A"n) of a quasi-Banach space X, such that A"1@?A"2@?...@?X, A"n+A"n@?A"K"("n"), and @?"nA"n@?=X. Continuing the line of research originating at the classical paper [8] by Bernstein, we give several characterizations of the approximation schemes with the property that, for every sequence {@e"n}@?0, there exists x@?X such that dist(x,A"n) O(@e"n) (in this case we say that (X,{A"n}) satisfies Shapiro's Theorem). If X is a Banach space, x@?X as above exists if and only if, for every sequence {@d"n}@?0, there exists y@?X such that dist(y,A"n)>=@d"n. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.
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