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.
@article{arxiv.1207.0946,
title = {Lebesgue type inequalities for quasi-greedy bases},
author = {Gustavo Garrigós and Eugenio Hernández and Timur Oikhberg},
journal= {arXiv preprint arXiv:1207.0946},
year = {2012}
}