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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 $A_1 \subsetneq A_2 \subsetneq ... \subsetneq X$, $A_n + A_n \subset A_{K(n)}$, and $\bar{\cup_n A_n} = X$. 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 $\{ε_n\}\searrow 0$, there exists $x\in X$ such that $dist(x,A_n)\neq \mathbf{O}(ε_n)$ (in this case we say that $(X,\{A_n\})$ satisfies Shapiro's Theorem). If $X$ is a Banach space, $x \in X$ as above exists if and only if, for every sequence $\{δ_n\} \searrow 0$, there exists $y \in X$ such that $dist(y,A_n) \geq δ_n$. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.