Lebesgue inequalities for the greedy algorithm in general bases

Berná, Pablo M.; Garrigós, Gustavo

doi:10.1007/s13163-017-0221-x

Cited by 42 publications

(69 citation statements)

References 20 publications

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“…The proof of this inequality is almost the same as Lemma 3.4 in [20]. It is clear that, for a quasi-greedy basis (e i ) with the quasi-greedy constant K, γ n ≤ g n ≤ K.…”

Section: Corollary 36 If Is a Quasi-greedy Basis Of A Banach Space Xmentioning

confidence: 76%

“…It is known from [20], Proposition 5.1, for this basis, that g n = 2n. So this is a non-quasi-greedy basis of X.…”

Section: It Is Clear That #(A) = #(B)mentioning

confidence: 95%

“…holds for any x ∈ X, n ≥ 1, while for TGA with respect to these bases, from Theorem 1.5 in [20], we know that, for any x ∈ X, n ≥ 1, the best inequality we can get is…”

Section: Corollary 36 If Is a Quasi-greedy Basis Of A Banach Space Xmentioning

confidence: 99%

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Lebesgue constants for Chebyshev thresholding greedy algorithms

Shao

2018

J Inequal Appl

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We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L ch n for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L ch n with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained. MSC: 41A25; 41A46; 42A10

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“…The proof of this inequality is almost the same as Lemma 3.4 in [20]. It is clear that, for a quasi-greedy basis (e i ) with the quasi-greedy constant K, γ n ≤ g n ≤ K.…”

Section: Corollary 36 If Is a Quasi-greedy Basis Of A Banach Space Xmentioning

confidence: 76%

“…It is known from [20], Proposition 5.1, for this basis, that g n = 2n. So this is a non-quasi-greedy basis of X.…”

Section: It Is Clear That #(A) = #(B)mentioning

confidence: 95%

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Lebesgue constants for Chebyshev thresholding greedy algorithms

Shao

2018

J Inequal Appl

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“…Of course, this operator is well defined since Λ α (x) is a finite set. In [4] we can find the following result:…”

Section: Preliminary Resultsmentioning

confidence: 92%

“…On the other hand, using the inequality (3.9) of [4], [4]. Given x ∈ X and CG t m ∈ G ch,t m , we denote by A = supp CG t m x ∈ G(x, m,t).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

et al. 2019

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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 [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper. 1991 Mathematics Subject Classification. 41A65, 41A46, 46B15.

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Characterization of Weight-Semi-greedy Bases

Berná

2020

J Fourier Anal Appl

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One classical result in greedy approximation theory is that almost-greedy and semigreedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by S. J. Dilworth, N. J. Kalton and D. Kutzarova in [7] and, recently, the first author in [2] proved that the condition of finite cotype can be removed in this result. In [11], the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and B is a Schauder basis in a Banach space X with finite cotype, then w-semi-greediness and w-almost-greediness are equivalent notions. In this paper, we prove the same characterization but removing the condition of finite cotype and, also, we try to relax the condition of Schauder in the characterization of semi-greediness using the ρ-admissibility, notion introduced recently in [4].2000 Mathematics Subject Classification. 46B15, 41A65.

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Lebesgue inequalities for the greedy algorithm in general bases

Cited by 42 publications

References 20 publications

Lebesgue constants for Chebyshev thresholding greedy algorithms

Lebesgue constants for Chebyshev thresholding greedy algorithms

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

Characterization of Weight-Semi-greedy Bases

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