# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Quasi-greedy bases and Lebesgue-type inequalities

### Tom 211 / 2012

Studia Mathematica 211 (2012), 41-69 MSC: Primary 41A65; Secondary 41A25, 41A46, 46B20. DOI: 10.4064/sm211-1-3

#### Streszczenie

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 $(\ln(m+1))^{1/2}$.

#### Autorzy

• S. J. DilworthDepartment of Mathematics
University of South Carolina
Columbia, SC 29208, U.S.A.
e-mail
• M. Soto-BajoDepartamento de Matemáticas
Cantoblanco, carretera de Colmenar Km 15