Wavelets on the interval 125
Moreover, for s ≥ 0, 1 p, q +∞, let
B
s pq,00
=
n v
∈ B
s pq
: supp v ⊆
o ;
3 then, if s
−
1 p
∈
✁
, B
s pq,00
= B
s pq,0
. 4
For these spaces, we have the real interpolation result: L
p
, B
s
1
pq,0
s
2
s
1
, p
= B
s
2
pq,00
, 5
where 1 p, q +∞ and 0 s
2
s
1
. In particular, if 1 p +∞, 0 s
2
s
1
, we have L
p
, W
s
1
, p
s
2
s
1
, p
=
W
s
2
, p
00
if s
2
−
1 p
∈
✁
, B
s
2
pq,0
if s
2
∈
✁
, W
s
2
, p
if s
2
, s
2
−
1 p
∈
✁
, 6
where W
s, p
, W
s, p 00
are defined similarly to the Besov spaces 2 and 3. We will also
consider negative values of s. For s 0, 1 p, q +∞, let us denote by p
′
and q
′
the conjugate index of p and q respectively i.e.
1 p
+
1 p
′
=
1 q
+
1 q
′
= 1. Then, we set B
s pq
=
B
−s p
′
q
′
,
′
.
2. Multilevel decompositions
We will recall how to define an abstract multilevel decomposition of a separable Banach space V , with norm denoted by
k·k, and how to obtain, using Jackson- and Bernstein-type inequalities, characterization of subspaces of V . For more details and proofs we refer, among the others, to
[5, 6, 15].
2.1. Abstract setting
Let {V
j
}
j ∈
✁
=
✂
or
✁
= { j ∈
✂
: j ≥ j
∈
✂
} be a family of closed subspaces of V such that V
j
⊂ V
j +1
, ∀ j ∈
✁
. For all j ∈
✁
, let P
j
: V → V
j
be a continuous linear operator satisfying the following properties:
kP
j
k
✄
V ,V
j
≤ C independent of j ,
7 P
j
v = v ,
∀v ∈ V
j
, 8
P
j
◦ P
j +1
= P
j
. 9
Observe that 7 and 8 imply kv − P
j
v k ≤ C inf
u∈V
j
kv − uk , ∀v ∈ V ,
where C is a constant independent of j . Through P
j
, we define another set of operators Q
j
: V
→ V
j +1
by Q
j
v = P
j +1
v − P
j
v , ∀v ∈ V , ∀ j ∈
✁
,
126 L. Levaggi – A. Tabacco
and the detail spaces W
j
: = Im Q
j
, ∀ j ∈
✁
. If
✁
is bounded from below, it will be convenient to set P
j −1
= 0 and V
j −1
= {0}; thus, Q
j −1
= P
j
and W
j −1
= V
j
. In this case, let us also set =
✁
∪ { j − 1}, otherwise
=
✁
. Thanks to the assumptions 7-8-9, every Q
j
is a continuous linear projection on W
j
, the sequence {Q
j
}
j ∈
is uniformly bounded in
✁
V , V
j +1
and each space W
j
is a complement space of V
j
in V
j +1
, i.e., V
j +1
= V
j
⊕ W
j
. 10
By iterating the decomposition 10, we get for any two integers j
1
, j
2
∈ such that j
1
j
2
V
j
2
= V
j
1
⊕
j
2
−1
M
j = j
1
W
j
,
11 so that every element in V
j
2
can be viewed as a rough approximation of itself on a coarse level plus a sum of refinement details. Making some more assumptions on the operators P
j
, we obtain a similar result for any v
∈ V . In fact, if P
j
v → v as
j → +∞ ,
12 and
P
j −1
= 0 if
✁
= { j ∈
✂
: j ≥ j
∈
✂
}, P
j
v → 0 as j → −∞ if
✁
=
✂
, 13
then V
=
+∞
M
j =inf
✂
W
j
, 14
and v
= X
j ≥inf
✂
Q
j
v , ∀v ∈ V .
15 The decomposition 15 is said to be q-stable, 1 q
∞, if kvk ≍
X
j ≥inf
✂
kQ
j
v k
q
1q
, ∀v ∈ V .
16 For each j
∈
✁
, let us fix a basis for the subspaces V
j
8
j
= n
ϕ
j k
: k ∈ ˘
✄
j
o ,
17 and for the subspaces W
j
9
j
= n
ψ
j k
: k ∈ ˆ
✄
j
o ,
18 with ˘
✄
j
and ˆ
✄
j
suitable sets of indices.
Wavelets on the interval 127
We can represent the operators P
j
and Q
j
in the form P
j
v =
X
k∈ ˘
j
˘v
j k
ϕ
j k
, Q
j
v =
X
k∈ ˆ
j
ˆv
j k
ψ
j k
, ∀v ∈ V .
19 Thus, if 12 and 13 hold, 15 can be rewritten as
v =
X
j ≥inf
✂
X
k∈ ˆ
j
ˆv
j k
ψ
j k
, ∀v ∈ V .
20 The bases chosen for the spaces V
j
and W
j
are called uniformly p-stable if, for a certain 1 p
∞, V
j
= X
k∈ ˘
j
α
k
ϕ
j k
: {α
k
}
k∈ ˘
j
∈ ℓ
p
and X
k∈ ˘
j
α
k
ϕ
j k
≍ {α
k
}
k∈ ˘
j
ℓ
p
, ∀ {α
k
} ∈ ℓ
p
, the constants involved in the definition of
≍ being independent of j. If the multilevel decomposition is q-stable, the bases 18 of each W
j
are uniformly p- stable, 12 and 13 hold, we can further transform 16 as
kvk ≍
X
j ≥inf
✂
X
k∈ ˆ
j
| ˆv
j k
|
p
q p
1q
, ∀v ∈ V .
2.2. Characterization of intermediate spaces
