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.

Abstract. We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.

We investigate various aspects of the "weighted" greedy algorithm with respect to a Schauder basis. For a weight w, we describe w-greedy, w-almost-greedy, and w-partiallygreedy bases, and examine some properties of w-semi-greedy bases. To achieve these goals, we introduce and study the w-Property (A).2000 Mathematics Subject Classification. 46B15, 41A65.

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