Characterization of 1-greedy bases

Abstract

A basis for a Banach space X is greedy if and only if the greedy algorithm provides, up to a constant C depending only on X , the best m -term approximation for each element of the space. It is known that the Haar (or good wavelet) basis is a greedy basis in L p ( 0 , 1 ) for 1 < p < ∞ [V.N. Temlyakov, The best m -term approximation and greedy algorithms, Adv. in Comp. Math. 8 (1998) 249–265]. In this particular example, unfortunately, the constant of greediness C = C ( p ) is strictly bigger than 1 unless p = 2 . Our goal is to investigate 1 -greedy bases, i.e., bases for which the greedy algorithm provides the best m -term approximation. We find a characterization of 1 -greediness, study how 1 -greedy bases relate to symmetric bases, and show that 1 -greediness does not imply 1 -symmetry, answering thus two questions raised in [P. Wojtaszczyk, Greedy Type Bases in Banach Spaces, Constructive Function Theory, Varna 2002, Darba, Sofia, 2002, pp. 1–20].