Shapiro's Theorem for subspaces

In the previous paper (Almira and Oikhberg. 2010 [4]), the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A(n)) (defined by E(x, A(n)) = inf(a is an element of An) parallel to x - a(n)parallel to) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive. (C) 2011 Elsevier Inc. All rights reserved.