We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasi-greedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of , 1 < p < ∞, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken as C(p)ln(m+1). The known magnitude of the corresponding multiplier for general (no assumption of uniform boundedness) quasi-greedy bases is of order , p ≠ 2. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing ln(m+1) by .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-1-3,
author = {S. J. Dilworth and M. Soto-Bajo and V. N. Temlyakov},
title = {Quasi-greedy bases and Lebesgue-type inequalities},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {41-69},
zbl = {1264.41032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-1-3}
}
S. J. Dilworth; M. Soto-Bajo; V. N. Temlyakov. Quasi-greedy bases and Lebesgue-type inequalities. Studia Mathematica, Tome 209 (2012) pp. 41-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-1-3/