Quasi-greedy bases and Lebesgue-type inequalities

Tom 211 / 2012

S. J. Dilworth, M. Soto-Bajo, V. N. Temlyakov 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
    Facultad de Ciencias
    Universidad Autónoma de Madrid
    Cantoblanco, carretera de Colmenar Km 15
    28049 Madrid, Spain
    e-mail
  • V. N. TemlyakovDepartment of Mathematics
    University of South Carolina
    Columbia, SC 29208, U.S.A.
    e-mail

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