Sparse approximation using new greedy-like bases in superreflexive spaces

TL;DR

This paper explores the theoretical optimality of new greedy-like bases in superreflexive Banach spaces, analyzing their efficiency in sparse approximation and comparing them to existing bases through unconditionality parameters.

Contribution

It introduces and studies new types of greedy-like bases, providing quantitative estimates and constructing bidemocratic bases with worse unconditionality parameters than almost greedy bases.

Findings

New greedy-like bases have worse unconditionality parameters than almost greedy bases.

Constructed bidemocratic bases demonstrate limitations in efficiency.

Enhanced methods allow for better understanding of basis optimality in Banach spaces.

Abstract

This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.