Quasi-greedy Bases and Lebesgue-type Inequalities

Abstract

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 quasigreedy 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 .