Approximation schemes satisfying Shapiro's Theorem

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 ∪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 {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.