Embeddings and Lebesgue-type inequalities for the greedy algorithm in Banach spaces

TL;DR

This paper establishes Lebesgue-type inequalities for the greedy algorithm in Banach spaces, linking bounds to democracy functions and embeddings, and demonstrates their asymptotic optimality through diverse examples.

Contribution

It introduces new Lebesgue-type inequalities for the greedy algorithm in Banach spaces based on democracy functions and embeddings, extending previous results.

Findings

Bounds depend only on democracy functions of basis and dual

Inequalities are equivalent to embeddings into weighted Lorentz spaces

Asymptotic optimality shown in examples with non-quasi-greedy bases

Abstract

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.