Quasi-greedy bases and Lebesgue-type inequalities

S. J. Dilworth, M. Soto-Bajo, and V. N. Temlyakov. "Quasi-greedy bases and Lebesgue-type inequalities." Studia Mathematica 211.1 (2012): 41-69. <http://eudml.org/doc/285514>.

@article{S2012,
abstract = {We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the $L_\{p\}$ 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 $L_\{p\}$, 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 $m^\{|1/2-1/p|\}$, p ≠ 2. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing ln(m+1) by $(ln(m+1))^\{1/2\}$.},
author = {S. J. Dilworth, M. Soto-Bajo, V. N. Temlyakov},
journal = {Studia Mathematica},
keywords = {greedy algorithms; quasi-greedy bases; Lebesgue-type inequalities},
language = {eng},
number = {1},
pages = {41-69},
title = {Quasi-greedy bases and Lebesgue-type inequalities},
url = {http://eudml.org/doc/285514},
volume = {211},
year = {2012},
}

TY - JOUR
AU - S. J. Dilworth
AU - M. Soto-Bajo
AU - V. N. Temlyakov
TI - Quasi-greedy bases and Lebesgue-type inequalities
JO - Studia Mathematica
PY - 2012
VL - 211
IS - 1
SP - 41
EP - 69
AB - We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the $L_{p}$ 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 $L_{p}$, 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 $m^{|1/2-1/p|}$, p ≠ 2. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing ln(m+1) by $(ln(m+1))^{1/2}$.
LA - eng
KW - greedy algorithms; quasi-greedy bases; Lebesgue-type inequalities
UR - http://eudml.org/doc/285514
ER -