Quasi-greedy bases and Lebesgue-type inequalities

• Autorzy: S. J. Dilworth , M. Soto-Bajo , V. N. Temlyakov
• Języki publikacji: EN

Abstrakty

EN

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

Kategorie tematyczne

• 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
• 46B20: Geometry and structure of normed linear spaces
• 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy
• 41A25: Rate of convergence, degree of approximation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 211
• Numer: 1
• Strony: 41-69

Daty

wydano

2012

Twórcy

autor

S. J. Dilworth

• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

autor

M. Soto-Bajo

• Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco, carretera de Colmenar Km 15, 28049 Madrid, Spain

autor

V. N. Temlyakov

• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

Identyfikatory

DOI

10.4064/sm211-1-3