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Quasi-greedy bases and Lebesgue-type inequalities

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Dilworth S., Soto-Bajo M., Temlyakov V.

Studia Mathematica

2012

tom 211

nr 1

41-69

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

Ograniczanie wyników

1 Studia Mathematica

1 Dilworth S.

1 Soto-Bajo M.

1 Temlyakov V.

1 2012

Wyniki wyszukiwania

Quasi-greedy bases and Lebesgue-type inequalities