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The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (xn, x * n ) ∞ n=1 is a biorthogonal system in X then for each x ∈ X we have a formal expansionxn. The thresholding greedy algorithm (with threshold ε > 0) applied to x is formally defined asxn. The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.

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.

Abstract. This article closes the cycle of characterizations of greedy-like bases in the "isometric" case initiated in [1] with the characterization of 1-greedy bases and continued in [2] with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight characterization of almost greedy bases.

It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k m [B]) ∞ m=1 of its conditionality constants verifies the estimate k m [B] = O(log m) and that if the reverse inequality log m = O(k m [B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k m [B] = O(log m) 1−ǫ for some ǫ > 0. However, in the existing literature one finds very few instances of spaces possessing quasigreedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [16] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k m [B] = O(log m) and superreflexive classical Banach spaces having for every ǫ > 0 quasi-greedy bases B with k m [B] = O(log m) 1−ǫ . Moreover, in most cases those bases will be almost greedy. 2010 Mathematics Subject Classification. 46B15, 41A65.