Approximation schemes satisfying Shapiro's Theorem

الأصل · EN

An approximation scheme is a family of homogeneous subsets (Aₙ) of a quasi-Banach space X, such that A₁ A₂... 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} 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} 0, there exists y ∈ X such that dist(y,Aₙ) ≥ δₙ. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.