The WCGA in $$L^p(\log L)^{\alpha }$$ Spaces
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $${{\mathbb {X}}}$$ X . First, we generalize Temlyakov’s theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus...
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| Vydáno v: | Constructive approximation Ročník 61; číslo 1; s. 115 - 147 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
01.02.2025
|
| ISSN: | 0176-4276, 1432-0940 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces
$${{\mathbb {X}}}$$
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
$${\texttt {A3}}$$
A
3
parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces
$${{\mathbb {X}}}=L^p(\log L)^{\alpha }$$
X
=
L
p
(
log
L
)
α
, with
$$1<p<\infty $$
1
<
p
<
∞
and
$${\alpha }\in {{\mathbb {R}}}$$
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
$$\phi (N)=O(N^{\max \{1,2/p'\}} \,(\log N)^{|{\alpha }| p'})$$
ϕ
(
N
)
=
O
(
N
max
{
1
,
2
/
p
′
}
(
log
N
)
|
α
|
p
′
)
. Moreover, this quantity is sharp when
$$p\le 2$$
p
≤
2
. Finally, an expression for
$$\phi (N)$$
ϕ
(
N
)
in the case of the trigonometric system is also given. |
|---|---|
| ISSN: | 0176-4276 1432-0940 |
| DOI: | 10.1007/s00365-023-09664-y |