Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces
For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants k m [ B ] verify the estimate k m [ B ] = O ( log m ) . Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-gre...
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| Vydané v: | Constructive approximation Ročník 49; číslo 1; s. 103 - 122 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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15.02.2019
Springer Nature B.V |
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| ISSN: | 0176-4276, 1432-0940 |
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| Abstract | For a conditional quasi-greedy basis
B
in a Banach space, the associated conditionality constants
k
m
[
B
]
verify the estimate
k
m
[
B
]
=
O
(
log
m
)
. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies
k
m
[
B
]
=
O
(
(
log
m
)
1
-
ϵ
)
for some
0
<
ϵ
<
1
, and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space
X
is not superreflexive, then there is a quasi-greedy basis
B
in a Banach space
Y
finitely representable in
X
with
k
m
[
B
]
≈
log
m
. As a consequence, we obtain that for every
2
<
q
<
∞
, there is a Banach space
X
of type 2 and cotype
q
possessing a quasi-greedy basis
B
with
k
m
[
B
]
≈
log
m
. We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence
B
with
k
m
[
B
]
≈
m
. |
|---|---|
| AbstractList | For a conditional quasi-greedy basis
B
in a Banach space, the associated conditionality constants
k
m
[
B
]
verify the estimate
k
m
[
B
]
=
O
(
log
m
)
. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies
k
m
[
B
]
=
O
(
(
log
m
)
1
-
ϵ
)
for some
0
<
ϵ
<
1
, and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space
X
is not superreflexive, then there is a quasi-greedy basis
B
in a Banach space
Y
finitely representable in
X
with
k
m
[
B
]
≈
log
m
. As a consequence, we obtain that for every
2
<
q
<
∞
, there is a Banach space
X
of type 2 and cotype
q
possessing a quasi-greedy basis
B
with
k
m
[
B
]
≈
log
m
. We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence
B
with
k
m
[
B
]
≈
m
. For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies km[B]=O((logm)1-ϵ) for some 0<ϵ<1, and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space X is not superreflexive, then there is a quasi-greedy basis B in a Banach space Y finitely representable in X with km[B]≈logm. As a consequence, we obtain that for every 2<q<∞, there is a Banach space X of type 2 and cotype q possessing a quasi-greedy basis B with km[B]≈logm. We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence B with km[B]≈m. |
| Author | Wojtaszczyk, Przemysław Ansorena, José L. Albiac, Fernando |
| Author_xml | – sequence: 1 givenname: Fernando surname: Albiac fullname: Albiac, Fernando email: fernando.albiac@unavarra.es organization: Mathematics Department, Universidad Pública de Navarra – sequence: 2 givenname: José L. surname: Ansorena fullname: Ansorena, José L. organization: Department of Mathematics and Computer Sciences, Universidad de La Rioja – sequence: 3 givenname: Przemysław surname: Wojtaszczyk fullname: Wojtaszczyk, Przemysław organization: Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Institute of Mathematics of the Polish Academy of Sciences |
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| Cites_doi | 10.1512/iumj.2014.63.5269 10.1007/s00365-013-9209-z 10.1016/j.jat.2015.08.006 10.2140/pjm.1972.41.409 10.1007/978-3-642-66557-8 10.4064/sm-58-1-45-90 10.1007/s10444-010-9155-2 10.4064/sm227-2-3 10.1006/jath.2000.3512 10.1007/s13163-016-0204-3 10.1007/978-3-319-31557-7 10.4064/sm159-1-4 10.1007/BFb0078146 10.1017/CBO9781316480588 10.1007/978-3-642-51633-7 |
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| Keywords | Cotype Reflexivity Finite representability Thresholding greedy algorithm Banach spaces Superreflexivity Conditional basis Type 46B15 Super property 41A65 Quasi-greedy basis Conditionality constants |
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| References | CR4 CR6 Bennett, Sharpley (CR5) 1988 Garrigós, Wojtaszczyk (CR8) 2014; 63 James (CR9) 1972; 41 CR15 CR14 Albiac, Ansorena (CR2) 2017; 30 Pisier (CR13) 2016 Temlyakov, Yang, Ye (CR17) 2011; 34 Maurey, Pisier (CR12) 1976; 58 Garrigós, Hernández, Oikhberg (CR7) 2013; 38 Konyagin, Temlyakov (CR10) 1999; 5 Wojtaszczyk (CR18) 2000; 107 Albiac, Ansorena (CR1) 2016; 201 Temlyakov, Yang, Ye (CR16) 2011; 17 Albiac, Ansorena, Garrigós, Hernández, Raja (CR3) 2015; 227 Lindenstrauss, Tzafriri (CR11) 1977 9399_CR15 VN Temlyakov (9399_CR16) 2011; 17 G Garrigós (9399_CR7) 2013; 38 9399_CR14 F Albiac (9399_CR2) 2017; 30 F Albiac (9399_CR1) 2016; 201 9399_CR4 RC James (9399_CR9) 1972; 41 9399_CR6 SV Konyagin (9399_CR10) 1999; 5 J Lindenstrauss (9399_CR11) 1977 P Wojtaszczyk (9399_CR18) 2000; 107 VN Temlyakov (9399_CR17) 2011; 34 G Garrigós (9399_CR8) 2014; 63 F Albiac (9399_CR3) 2015; 227 C Bennett (9399_CR5) 1988 B Maurey (9399_CR12) 1976; 58 G Pisier (9399_CR13) 2016 |
| References_xml | – volume: 63 start-page: 1017 issue: 4 year: 2014 end-page: 1036 ident: CR8 article-title: Conditional quasi-greedy bases in hilbert and banach spaces publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2014.63.5269 – volume: 38 start-page: 447 issue: 3 year: 2013 end-page: 470 ident: CR7 article-title: Lebesgue-type inequalities for quasi-greedy bases publication-title: Constr. Approx. doi: 10.1007/s00365-013-9209-z – volume: 201 start-page: 7 year: 2016 end-page: 12 ident: CR1 article-title: Characterization of 1-quasi-greedy bases publication-title: J. Approx. Theory doi: 10.1016/j.jat.2015.08.006 – ident: CR4 – volume: 41 start-page: 409 year: 1972 end-page: 419 ident: CR9 article-title: Super-reflexive spaces with bases publication-title: Pac. J. Math. doi: 10.2140/pjm.1972.41.409 – ident: CR14 – ident: CR15 – year: 1977 ident: CR11 publication-title: Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete doi: 10.1007/978-3-642-66557-8 – volume: 58 start-page: 45 issue: 1 year: 1976 end-page: 90 ident: CR12 article-title: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de banach (in French) publication-title: Studia Math. doi: 10.4064/sm-58-1-45-90 – volume: 34 start-page: 319 issue: 3 year: 2011 end-page: 337 ident: CR17 article-title: Greedy approximation with regard to non-greedy bases publication-title: Adv. Comput. Math. doi: 10.1007/s10444-010-9155-2 – volume: 227 start-page: 133 issue: 2 year: 2015 end-page: 140 ident: CR3 article-title: Conditionality constants of quasi-greedy bases in super-reflexive Banach spaces publication-title: Studia Math. doi: 10.4064/sm227-2-3 – year: 1988 ident: CR5 publication-title: Interpolation of Operators, Pure and Applied Mathematics – ident: CR6 – volume: 17 start-page: 203 issue: 2 year: 2011 end-page: 214 ident: CR16 article-title: Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases publication-title: East J. Approx. – volume: 107 start-page: 293 issue: 2 year: 2000 end-page: 314 ident: CR18 article-title: Greedy algorithm for general biorthogonal systems publication-title: J. Approx. Theory doi: 10.1006/jath.2000.3512 – volume: 30 start-page: 13 issue: 1 year: 2017 end-page: 24 ident: CR2 article-title: Characterization of 1-almost greedy bases publication-title: Rev. Mat. Complut. doi: 10.1007/s13163-016-0204-3 – volume: 5 start-page: 365 issue: 3 year: 1999 end-page: 379 ident: CR10 article-title: A remark on greedy approximation in banach spaces publication-title: East J. Approx. – year: 2016 ident: CR13 publication-title: Martingales in Banach Spaces – volume: 5 start-page: 365 issue: 3 year: 1999 ident: 9399_CR10 publication-title: East J. Approx. – volume: 107 start-page: 293 issue: 2 year: 2000 ident: 9399_CR18 publication-title: J. Approx. Theory doi: 10.1006/jath.2000.3512 – volume: 38 start-page: 447 issue: 3 year: 2013 ident: 9399_CR7 publication-title: Constr. Approx. doi: 10.1007/s00365-013-9209-z – volume-title: Interpolation of Operators, Pure and Applied Mathematics year: 1988 ident: 9399_CR5 – volume: 63 start-page: 1017 issue: 4 year: 2014 ident: 9399_CR8 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2014.63.5269 – ident: 9399_CR4 doi: 10.1007/978-3-319-31557-7 – volume: 30 start-page: 13 issue: 1 year: 2017 ident: 9399_CR2 publication-title: Rev. Mat. Complut. doi: 10.1007/s13163-016-0204-3 – volume: 227 start-page: 133 issue: 2 year: 2015 ident: 9399_CR3 publication-title: Studia Math. doi: 10.4064/sm227-2-3 – ident: 9399_CR6 doi: 10.4064/sm159-1-4 – volume: 41 start-page: 409 year: 1972 ident: 9399_CR9 publication-title: Pac. J. Math. doi: 10.2140/pjm.1972.41.409 – volume-title: Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete year: 1977 ident: 9399_CR11 doi: 10.1007/978-3-642-66557-8 – ident: 9399_CR14 doi: 10.1007/BFb0078146 – volume-title: Martingales in Banach Spaces year: 2016 ident: 9399_CR13 doi: 10.1017/CBO9781316480588 – volume: 17 start-page: 203 issue: 2 year: 2011 ident: 9399_CR16 publication-title: East J. Approx. – ident: 9399_CR15 doi: 10.1007/978-3-642-51633-7 – volume: 201 start-page: 7 year: 2016 ident: 9399_CR1 publication-title: J. Approx. Theory doi: 10.1016/j.jat.2015.08.006 – volume: 58 start-page: 45 issue: 1 year: 1976 ident: 9399_CR12 publication-title: Studia Math. doi: 10.4064/sm-58-1-45-90 – volume: 34 start-page: 319 issue: 3 year: 2011 ident: 9399_CR17 publication-title: Adv. Comput. Math. doi: 10.1007/s10444-010-9155-2 |
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| Snippet | For a conditional quasi-greedy basis
B
in a Banach space, the associated conditionality constants
k
m
[
B
]
verify the estimate
k
m
[
B
]
=
O
(
log
m
)
.... For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). Answering a question... |
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| SubjectTerms | Analysis Banach spaces Mathematics Mathematics and Statistics Numerical Analysis |
| Title | Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces |
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