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    Asymptotic greediness of the Haar system in the spaces Lp[0 , 1] , 1< p< ∞

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    2019060033_Albiar_AsymptoticGreediness.pdf (365.8Kb)
    Date
    2019
    Author
    Albiac Alesanco, Fernando José Upna
    Ansorena, José L. 
    Berná, Pablo M. 
    Version
    Acceso abierto / Sarbide irekia
    xmlui.dri2xhtml.METS-1.0.item-type
    Artículo / Artikulua
    Version
    Versión aceptada / Onetsi den bertsioa
    Project Identifier
    ES/1PE/MTM2014-53009-P 
    ES/1PE/MTM2016-76808-P 
    ES/1PE/MTM2016-76566-P 
    Impact
     
     
     
    10.1007/s00365-019-09466-1
     
     
     
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    Abstract
    Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant Cg[H(p), Lp] of the (normalized) Haar system H(p) in Lp[0 , 1] for 1 < p < ∞. We will show that the super-democracy constant of H(p) in Lp[0 , 1] grows as p∗= max { p, p/ (p- 1) } as p∗ goes to ∞. Thus, since the unconditionality constant of H(p) in Lp[0 , 1] is p∗- 1 , the well-known general estimates for the ... [++]
    Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant Cg[H(p), Lp] of the (normalized) Haar system H(p) in Lp[0 , 1] for 1 < p < ∞. We will show that the super-democracy constant of H(p) in Lp[0 , 1] grows as p∗= max { p, p/ (p- 1) } as p∗ goes to ∞. Thus, since the unconditionality constant of H(p) in Lp[0 , 1] is p∗- 1 , the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that p∗≲Cg[H(p),Lp]≲(p∗)2. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, Cg[H(p), Lp] ≈ p∗. Our work answers a question that was raised by Hytonen (2015). [--]
    Subject
    Haar basis, Greedy basis, Unconditional basis, Democratic basis, Lp spaces, Lebesgue-type inequalities
     
    Publisher
    Springer
    Published in
    Constructive Approximation, 2019
    Departament
    Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas / Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila / Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa. InaMat - Institute for Advanced Materials
     
    Publisher version
    https://doi.org/10.1007/s00365-019-09466-1
    URI
    https://hdl.handle.net/2454/35932
    Sponsorship
    F. Albiac and J. L. Ansorena acknowledge the support of the grant MTM2014-53009-P (MINECO, Spain). F. Albiac was also supported by the grant MTM2016-76808-P (MINECO, Spain). P. M. Bern a was supported by a Ph.D. fellowship from the program FPI-UAM, as well as the grants MTM-2016-76566-P (MINECO, Spain) and 19368/PI/14 (Fundación Séneca, Región de Murcia, Spain).
    Appears in Collections
    • Artículos de revista - Aldizkari artikuluak [2924]
    • Artículos de revista InaMat - InaMat aldizkari artikuluak [142]
    • Artículos de revista DEIM - EIMS Aldizkari artikuluak [103]
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