Characterization of Weight-Semi-greedy Bases

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math A...

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Vydané v:	The Journal of fourier analysis and applications Ročník 26; číslo 1
Hlavný autor:	Berná, Pablo Manuel
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.02.2020 Springer Springer Nature B.V
Predmet:	Abstract Harmonic Analysis Approximations and Expansions Banach spaces Democracy Equivalence Fourier Analysis Mathematical Methods in Physics Mathematics Mathematics and Statistics Partial Differential Equations Signal,Image and Speech Processing Weight Weight-almost-greedy bases 46B15 Thresholding greedy algorithm Semi-greedy bases 41A65
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Abstract	One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the ρ -admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut; https://doi.org/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120–141, 2018), 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. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w -greedy-type bases and some w -democracy properties.
AbstractList	One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the ρ -admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut; https://doi.org/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120–141, 2018), 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. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w -greedy-type bases and some w -democracy properties. One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the ρ-admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut; https://doi.org/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120–141, 2018), 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. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w-greedy-type bases and some w-democracy properties. One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67-101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218-225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the [Formula omitted]-admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut;
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References	Dilworth, Kalton, Kutzarova, Temlyakov (CR8) 2003; 19 CR4 Oikhberg (CR14) 2018; 255 Wojtaszczyk (CR17) 2000; 107 Albiac, Ansorena (CR1) 2017; 30 Temlyakov (CR16) 2015 Berná, Dilworth, Kutzarova, Oikhberg, Wallis (CR5) 2019; 248 Dilworth, Kutzarova, Temlyakov (CR10) 2002; 8 Cohen, DeVore, Hochmuth (CR6) 2000; 16 Temlyakov (CR15) 2011 Berná (CR2) 2019; 470 Dilworth, Kutzarova, Oikhberg (CR9) 2015; 28 Dilworth, Kalton, Kutzarova (CR7) 2003; 159 Dilworth, Kutzarova, Temlyakov, Wallis (CR11) 2018; 303 Konyagin, Temlyakov (CR13) 1999; 5 Kerkyacharian, Picard, Temlyakov (CR12) 2006; 12 Berná, Blasco, Garrigós (CR3) 2017; 30 SJ Dilworth (9727_CR7) 2003; 159 VN Temlyakov (9727_CR15) 2011 T Oikhberg (9727_CR14) 2018; 255 PM Berná (9727_CR2) 2019; 470 SJ Dilworth (9727_CR8) 2003; 19 G Kerkyacharian (9727_CR12) 2006; 12 SJ Dilworth (9727_CR9) 2015; 28 PM Berná (9727_CR3) 2017; 30 PM Berná (9727_CR5) 2019; 248 P Wojtaszczyk (9727_CR17) 2000; 107 F Albiac (9727_CR1) 2017; 30 SJ Dilworth (9727_CR11) 2018; 303 SJ Dilworth (9727_CR10) 2002; 8 9727_CR4 SV Konyagin (9727_CR13) 1999; 5 A Cohen (9727_CR6) 2000; 16 VN Temlyakov (9727_CR16) 2015
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SubjectTerms	Abstract Harmonic Analysis Approximations and Expansions Banach spaces Democracy Equivalence Fourier Analysis Mathematical Methods in Physics Mathematics Mathematics and Statistics Partial Differential Equations Signal,Image and Speech Processing Weight
Title	Characterization of Weight-Semi-greedy Bases
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