The WCGA in Lp(logL)α Spaces
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces X . First, we generalize Temlyakov’s theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus of smoothness and th...
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| Veröffentlicht in: | Constructive approximation Jg. 61; H. 1; S. 115 - 147 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.02.2025
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0176-4276, 1432-0940 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces
X
. First, we generalize Temlyakov’s theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus of smoothness and the
A
3
parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces
X
=
L
p
(
log
L
)
α
, with
1
<
p
<
∞
and
α
∈
R
, and show that, when the Haar system is used, then exact recovery of
N
-sparse signals occurs when the number of iterations is
ϕ
(
N
)
=
O
(
N
max
{
1
,
2
/
p
′
}
(
log
N
)
|
α
|
p
′
)
. Moreover, this quantity is sharp when
p
≤
2
. Finally, an expression for
ϕ
(
N
)
in the case of the trigonometric system is also given. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0176-4276 1432-0940 |
| DOI: | 10.1007/s00365-023-09664-y |