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 error to the best N-term approximation error with bounds depending on democracy functions and the basis's properties.

Contribution

It provides new bounds for the thresholding greedy algorithm error in Banach spaces, especially for democratic bases, with explicit dependence on basis properties.

Findings

Error bounds depend on democracy functions and quasi-greedy constants.

For democratic bases, the error bound is proportional to log N.

Examples show the bounds are sharp for democratic bases.

Abstract

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases.