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 (Formula Presented) is a biorthogonal system in X then for each x ∈ X we have a formal expansion (Formula Presented). The thresholding greedy algorithm (with threshold e > 0) applied to x is formally defined as (Formula Presented). 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.

Greedy approximation for biorthogonal systems in quasi-Banach spaces

Non-autonomous nonlocal partial differential equations with delay and memory

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 not necessarily quasi-greedy bases.

Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems (also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations amongst them are carefully discussed.

Greedy approximation for biorthogonal systems in quasi-Banach spaces.

Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

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

Pablo M. Berná

Papers

Greedy approximation for biorthogonal systems in quasi-Banach spaces

Lebesgue inequalities for the greedy algorithm in general bases

Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

Greedy approximation for biorthogonal systems in quasi-Banach spaces.

Characterization of greedy bases in Banach spaces