Approximation schemes satisfying Shapiro's Theorem

An approximation scheme is a family of homogeneous subsets (A _n) of a quasi-Banach space X, such that A ₁{subset of with not equal to}A ₂{subset of with not equal to}...{subset of with not equal to}X, A _n+A _n⊂A _K(n), and ∪nAn-=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 {ε _n}↘0, there exists x∈X such that dist(x, A _n)≠O(ε _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 {δ _n}↘0, there exists y∈X such that dist(y, A _n)≥δ _n. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.