## Quasi-greedy bases and Lebesgue-type inequalities

**A 2012 Preprint by
S. Dilworth,
M. Soto-Bajo, and
V. Temlyakov
**

- 2012:02
We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on this study in 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 $m^{|\frac{1}{2}-\frac{1}{p}|}$, $p\neq 2$. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing $\ln(m+1)$ by $(\ln(m+1))^{1/2}$.