Quasi-greedy bases and Lebesgue-type inequalities

S. J. Dilworth ¹   ; M. Soto-Bajo ²   ; V. N. Temlyakov ¹

¹ Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.
² Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid Cantoblanco, carretera de Colmenar Km 15 28049 Madrid, Spain

Studia Mathematica, Tome 211 (2012) no. 1, pp. 41-69

Résumé

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 $L_p$, $1 p \infty$, 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 $m^{|1/2-1/p|}$, $p\neq 2$. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing $\ln(m+1)$ by $(\ln(m+1))^{1/2}$.

Détail

DOI : 10.4064/sm211-1-3

Keywords: study lebesgue type inequalities greedy approximation respect quasi greedy bases mostly concentrate spaces novelty paper obtaining better lebesgue type inequalities under extra assumptions quasi greedy basis known lebesgue type inequalities quasi greedy bases consider uniformly bounded quasi greedy bases infty prove bases extra multiplier lebesgue type inequality taken known magnitude corresponding multiplier general assumption uniform boundedness quasi greedy bases order neq uniformly bounded orthonormal quasi greedy bases get further improvements replacing