Maximal inequalities, frames and greedy algorithms

TL;DR

This paper explores the relationship between maximal inequalities, frames, and greedy algorithms in Banach lattices, establishing conditions for order convergence and characterizing the geometry of function spaces.

Contribution

It introduces new maximal inequalities for coordinate systems, analyzes the convergence of greedy algorithms, and characterizes Banach lattices via unconditional sequences.

Findings

Greedy algorithm convergence is equivalent to a specific maximal inequality.

Absolute frames may not admit usual order reconstruction but do so in the double dual.

Banach lattice is isomorphic to a sublattice of C(K) iff every unconditional sequence is absolute.

Abstract

The aim of this article is to use Banach lattice techniques to study coordinate systems in function spaces. We begin by proving that the greedy algorithm of a basis is order convergent if and only if a certain maximal inequality is satisfied. We then show that absolute frames need not admit a reconstruction algorithm with respect to the usual order convergence, but do allow for reconstruction with respect to the order convergence inherited from the double dual. After this, we investigate the extent to which such coordinate systems affect the geometry of the underlying function space. Most notably, we prove that a Banach lattice $X$ is lattice isomorphic to a closed sublattice of a $C(K)$-space if and only if every unconditional sequence in $X$ is absolute.