Shapiro's Theorem for subspaces

TL;DR

This paper explores the conditions under which elements with arbitrarily slow approximation errors can be chosen within specific subspaces of quasi-Banach spaces, extending previous results on Shapiro's theorem.

Contribution

It generalizes the existence of elements with slow approximation rates to be confined within prescribed subspaces in quasi-Banach spaces.

Findings

Existence of elements with slow approximation in specific subspaces in many cases

Extension of Shapiro's theorem to subspace selection

Conditions under which the selection is possible

Abstract

In a previous paper (see arXiv:1003.3411 [math.CA]), we 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 \in A_n} \|x - a_n\|) 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.