Lebesgue inequalities for Chebyshev thresholding greedy algorithms
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Lebesgue inequalities for Chebyshev thresholding greedy algorithms P. M. Berná1
· Ó. Blasco2 · G. Garrigós3 · E. Hernández1 · T. Oikhberg4
Received: 11 November 2018 / Accepted: 10 September 2019 © Universidad Complutense de Madrid 2019
Abstract We establish estimates for the Lebesgue parameters of the Chebyshev weak thresholding greedy algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in Dilworth et al. (Rev Mat Complut 28(2):393– 409, 2015), and are complemented with examples showing the optimality of the bounds. Our results also clarify certain bounds recently announced in Shao and Ye (J Inequal Appl 2018(1):102, 2018), and answer some questions left open in that paper. Keywords Thresholding Chebyshev greedy algorithm · Thresholding greedy algorithm · Quasi-greedy basis · Semi-greedy bases Mathematics Subject Classification 41A65 · 41A46 · 46B15
1 Introduction Let X be a Banach space over K = R or C, let X∗ be its dual space, and consider a ∗ system {en , en∗ }∞ n=1 ⊂ X × X with the following properties: ∗ (a) 0 < inf n {en , en } ≤ supn {en , en∗ } < ∞ (b) en∗ (em ) = δn,m , for all n, m ≥ 1 (c) X = span {en : n ∈ N}
The research of the authors P. M. Berná, G. Garrigós and E. Hernández are partially supported by the Grants MTM-2016-76566-P (MINECO, Spain) and 19368/PI/14 (Fundación Séneca, Región de Murcia, Spain). Also, P. M. Berná was supported by a Ph.D fellowship of the program “Ayudas para contratos predoctorales para la formación de doctores 2017” (MINECO, Spain). Ó. Blasco is supported by Grant MTM-2014-53009-P (MINECO, Spain). G. Garrigós partially supported by Grant MTM2017-83262-C2-2-P (Spain). E. Hernández has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 777822.
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P. M. Berná [email protected]
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P. M. Berná et al. w∗
(d) X∗ = span {en∗ : n ∈ N} . Under these conditions B = {en }∞ n=1 is called a seminormalized Markushevich basis for X (or M-basis for short), with dual system {en∗ }∞ n=1 . Sometimes we shall consider the following special cases N (e) B is a Schauder basis if K b := sup N S N < ∞, where S N x := n=1 en∗ (x)en is the N -th partial sum operator N Sn is the N -th (f) B is a Cesàro basis if sup N FN < ∞, where FN := N1 n=1 (C,1)-Cesàro operator. In this case we use the constant β = max sup FN , sup I − FN . N
(1.1)
N
For the latter terminology, see e.g. [21, Def. III.11.1]. With every x ∈ X, we shall ∗ ∗ associate the formal series x ∼ ∞ n=1 en (x)en , where a)-c) imply that lim n en (x) = 0. ∗ As usual, we denote suppx = {n ∈ N : en (x) = 0}. We recall standard notions about (weak) greedy algorithms; see e.g. the texts [23,25] for details and historical background. Fix t ∈ (0, 1]. We say that A is a t-greedy set for x of order m, denoted A ∈ G(x, m, t), if |A| = m and min |en∗ (x)| ≥ t · max |en∗ (x)|. n ∈A /
n∈A
(1.2)
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