Lebesgue-Type Inequalities for Quasi-greedy Bases
We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N -term approximation ∥ x − G N x ∥ and the best N -term approximation σ N ( x ) is controlled by max{ μ ( N ), k N }, where μ ( N ) and k N are well-known constants that quantify the...
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| Vydáno v: | Constructive approximation Ročník 38; číslo 3; s. 447 - 470 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Boston
Springer US
01.12.2013
|
| Témata: | |
| ISSN: | 0176-4276, 1432-0940 |
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| Abstract | We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy
N
-term approximation ∥
x
−
G
N
x
∥ and the best
N
-term approximation
σ
N
(
x
) is controlled by max{
μ
(
N
),
k
N
}, where
μ
(
N
) and
k
N
are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is
O
(log
N
). We show with various examples that these bounds are actually attained. |
|---|---|
| AbstractList | We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy
N
-term approximation ∥
x
−
G
N
x
∥ and the best
N
-term approximation
σ
N
(
x
) is controlled by max{
μ
(
N
),
k
N
}, where
μ
(
N
) and
k
N
are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is
O
(log
N
). We show with various examples that these bounds are actually attained. |
| Author | Hernández, Eugenio Oikhberg, Timur Garrigós, Gustavo |
| Author_xml | – sequence: 1 givenname: Gustavo surname: Garrigós fullname: Garrigós, Gustavo organization: Departamento de Matemáticas, Universidad de Murcia – sequence: 2 givenname: Eugenio surname: Hernández fullname: Hernández, Eugenio email: eugenio.hernandez@uam.es organization: Departamento de Matemáticas, Universidad Autónoma de Madrid – sequence: 3 givenname: Timur surname: Oikhberg fullname: Oikhberg, Timur organization: Department of Mathematics, University of Illinois |
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| Cites_doi | 10.4064/sm144-1-4 10.1007/s10444-011-9197-0 10.1006/jath.2000.3512 10.1112/S0024611503014084 10.1007/s10444-010-9155-2 10.1017/CBO9780511762291 10.1007/s00365-002-0525-y 10.1007/978-3-642-66557-8 10.1016/j.acha.2007.06.001 10.1353/ajm.1999.0016 10.4064/sm161-3-1 10.4064/sm159-1-4 10.1007/s003659900090 10.4064/ba53-1-4 10.4064/sm211-1-3 10.1007/978-3-642-51633-7 10.1017/S0962492906380014 10.4171/RMI/345 10.1512/iumj.2014.63.5269 |
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| Issue | 3 |
| Keywords | 41A46 46B15 Non-linear approximation Bounded variation 41A65 Thresholding greedy algorithm Quasi-greedy bases Lebesgue-type inequalities Democracy functions 41A17 |
| Language | English |
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Approx. doi: 10.1007/s00365-002-0525-y – volume: 34 start-page: 319 year: 2011 ident: 9209_CR22 publication-title: Adv. Comput. Math. doi: 10.1007/s10444-010-9155-2 – volume: 121 start-page: 587 issue: 3 year: 1999 ident: 9209_CR3 publication-title: Am. J. Math. doi: 10.1353/ajm.1999.0016 – volume: 28 start-page: 305 issue: 4 year: 2002 ident: 9209_CR15 publication-title: Serdica Math. J. – volume: 12 start-page: 539 year: 1964 ident: 9209_CR16 publication-title: Bull. Acad. Polon. Sci. – volume-title: Classical Banach Spaces, Vol I year: 1977 ident: 9209_CR17 doi: 10.1007/978-3-642-66557-8 – volume: 144 start-page: 95 year: 2001 ident: 9209_CR7 publication-title: Stud. Math. doi: 10.4064/sm144-1-4 – volume-title: Greedy Approximation year: 2011 ident: 9209_CR21 doi: 10.1017/CBO9780511762291 |
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| Snippet | We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy
N
-term approximation ∥
x
−
G
N
x
∥ and... |
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| SubjectTerms | Analysis Mathematics Mathematics and Statistics Numerical Analysis |
| Title | Lebesgue-Type Inequalities for Quasi-greedy Bases |
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