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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| Vydané v: | Constructive approximation Ročník 61; číslo 1; s. 115 - 147 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.02.2025
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0176-4276, 1432-0940 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 A3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces X=Lp(logL)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(Nmax{1,2/p′}(logN)|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. |
| Author | Garrigós, Gustavo |
| Author_xml | – sequence: 1 givenname: Gustavo surname: Garrigós fullname: Garrigós, Gustavo email: gustavo.garrigos@um.es organization: Departamento de Matemáticas, Universidad de Murcia |
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| Copyright | The Author(s) 2023 The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Snippet | We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces
X
. First,... We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces X. First, we... |
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| SubjectTerms | Algorithms Analysis Approximation Banach spaces Chebyshev approximation Dictionaries Greedy algorithms Mathematics Mathematics and Statistics Numerical Analysis Smoothness Theorems |
| Title | The WCGA in Lp(logL)α Spaces |
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