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Schreier families and
F
\mathcal{F}
F
-(almost) greedy bases
Authors
Kevin Beanland
Hung Viet Chu
Publication date
2 November 2022
Publisher
View
on
arXiv
Abstract
Let
F
\mathcal{F}
F
be a hereditary collection of finite subsets of
N
\mathbb{N}
N
. In this paper, we introduce and characterize
F
\mathcal{F}
F
-(almost) greedy bases. Given such a family
F
\mathcal{F}
F
, a basis
(
e
n
)
n
(e_n)_n
(
e
n
β
)
n
β
for a Banach space
X
X
X
is called
F
\mathcal{F}
F
-greedy if there is a constant
C
β©Ύ
1
C\geqslant 1
C
β©Ύ
1
such that for each
x
β
X
x\in X
x
β
X
,
m
β
N
m \in \mathbb{N}
m
β
N
, and
G
m
(
x
)
G_m(x)
G
m
β
(
x
)
, we have
β₯
x
β
G
m
(
x
)
β₯
Β
β©½
Β
C
inf
β‘
{
β₯
x
β
β
n
β
A
a
n
e
n
β₯
β
:
β
β£
A
β£
β©½
m
,
A
β
F
,
(
a
n
)
β
K
}
.
\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.
β₯
x
β
G
m
β
(
x
)
β₯
Β
β©½
Β
C
in
f
{
β
x
β
n
β
A
β
β
a
n
β
e
n
β
β
:
β£
A
β£
β©½
m
,
A
β
F
,
(
a
n
β
)
β
K
}
.
Here
G
m
(
x
)
G_m(x)
G
m
β
(
x
)
is a greedy sum of
x
x
x
of order
m
m
m
, and
K
\mathbb{K}
K
is the scalar field. From the definition, any
F
\mathcal{F}
F
-greedy basis is quasi-greedy and so, the notion of being
F
\mathcal{F}
F
-greedy lies between being greedy and being quasi-greedy. We characterize
F
\mathcal{F}
F
-greedy bases as being
F
\mathcal{F}
F
-unconditional,
F
\mathcal{F}
F
-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for
F
\mathcal{F}
F
-almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially
F
\mathcal{F}
F
-greedy. For a countable ordinal
Ξ±
\alpha
Ξ±
, we consider the case
F
=
S
Ξ±
\mathcal{F}=\mathcal{S}_\alpha
F
=
S
Ξ±
β
, where
S
Ξ±
\mathcal{S}_\alpha
S
Ξ±
β
is the Schreier family of order
Ξ±
\alpha
Ξ±
. We show that for each
Ξ±
\alpha
Ξ±
, there is a basis that is
S
Ξ±
\mathcal{S}_{\alpha}
S
Ξ±
β
-greedy but is not
S
Ξ±
+
1
\mathcal{S}_{\alpha+1}
S
Ξ±
+
1
β
-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals
Ξ±
<
Ξ²
\alpha < \beta
Ξ±
<
Ξ²
,
\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$
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Last time updated on 08/12/2022