Lebesgue-Type inequalities for quasi-greedy bases

TL;DR

This paper establishes Lebesgue-type inequalities for quasi-greedy bases in Banach spaces, linking the greedy algorithm's error to the best N-term approximation error with bounds depending on basis properties.

Contribution

It provides new bounds for the thresholding greedy algorithm's error in terms of democracy functions and quasi-greedy constants, advancing understanding of approximation efficiency.

Findings

Error of greedy algorithm bounded by best N-term approximation error

Bounds depend on democracy functions and quasi-greedy constant

Improves theoretical understanding of greedy approximation in Banach spaces

Abstract

We show that for quasi-greedy bases in real Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a constant which depends on the democracy functions and the quasi-greedy constant of the basis.