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
Let us consider a Banach space Z ⊂ V , whose norm will be denoted by k · k
Z
. We assume that there exists a semi-norm
| · |
Z
in Z such that kvk
Z
≍ kvk + |v|
Z
, ∀v ∈ Z .
21 In addition, we assume that
V
j
⊂ Z , ∀ j ∈
✁
. 22
Thus, Z is included in V with continuous embedding. We recall that the real interpolation method [4] allows us to define a family of intermediate
spaces Z
α q
, with 0 α 1 and 1 q ∞, such that
Z ⊂ Z
α
2
q
2
⊂ Z
α
1
q
1
⊂ V , 0 α
1
α
2
1 , 1 q
1
, q
2
∞ , with continuous inclusion. The space Z
α q
is defined as Z
α q
= V, Z
α, q
= v
∈ V : |v|
q α,
q
: =
Z
∞
[t
−α
K v, t ]
q
dt t
∞ ,
128 L. Levaggi – A. Tabacco
where K v, t
= inf
z∈Z
{kv − zk + t |z|
Z
} , v
∈ V, t 0 . Z
α q
is equipped with the norm kvk
α, q
= kvk
q
+ |v|
q α,
q 1q
. We note that one can replace the semi-norm
|v|
α, q
by an equivalent, discrete version as follows: |v|
α, q
≍
X
j ∈
b
α q j
K v, b
− j q
1q
, ∀v ∈ V ,
where b is any real number 1. We can characterize the space Z
α q
in terms of the multilevel decomposition introduced in the previous Subsection see [5, 6] and also [20]. This general result is based on two inequalities
classically known as Bernstein and Jackson inequalities. In our framework, these inequalities read as follows: there exists a constant b 1 such that the Bernstein inequality
|v|
Z
b
j
kvk , ∀v ∈ V
j
, ∀ j ∈
✁
23 and the Jackson inequality
kv − P
j
v k
b
− j
|v|
Z
, ∀v ∈ Z ,
∀ j ∈
✁
24 hold. The Bernstein inequality is also known as an inverse inequality, since it allows the stronger
norm kvk
Z
to be bounded by the weaker norm kvk, provided v ∈ V
j
. The Jackson inequality is an approximation result, which yields the rate of decay of the approximation error by P
j
for an element belonging to Z . Note that, if we assume Z to be dense in V , then the Jackson inequality
implies the consistency condition 12. The following characterization theorem holds. T
HEOREM
1. Let {V
j
, P
j
}
j ∈
be a family as described in Section 2.1. Let Z be a subspace of V satisfying the hypotheses 21 and 22. If the Bernstein and Jackson inequalities 23 and
24 hold, then for all 0 α 1, 1 q ∞ one has
Z
α q
=
v ∈ V :
X
j ≥inf
✂
b
α q j
kQ
j
v k
q
+∞
with |v|
α, q
≍
X
j ≥inf
✂
b
α q j
kQ
j
v k
q
1q
, ∀v ∈ Z
α q
. If in addition both 12 and 13 are satisfied, the bases 18 of each W
j
j ∈
✁
are uniformly p-stable for suitable 1 p
∞ and the multilevel decomposition 15 is q-stable, then the following representation of the norm of Z
α q
holds:
kvk
α, q
≍
X
j ∈
1 + b
α q j
X
k∈ ˆ
j
| ˆv
j k
|
p
q p
1q
, ∀v ∈ Z
α q
, if
✁
=
✂
,
Wavelets on the interval 129
or kvk
α, q
≍ kP
j
v k +
X
j ∈
b
α q j
X
k∈ ˆ
j
| ˆv
j k
|
p
q p
1q
, ∀v ∈ Z
α q
, if
✁
= { j ∈
✂
: j ≥ j
∈
✂
}. It is possible to prove a similar result for dual topological spaces see e.g. [5]. Indeed, let V
be reflexive and h f, vi denote the dual pairing between the topological dual space V
′
and V . For each j
∈
✁
, let e P
j
: V
′
→ V
′
be the adjoint operator of P
j
and e V
j
= Im e P
j
⊂ V
′
. It is possible to show that the family
{e V
j
, e P
j
} satisfies the same abstract assumptions of {V
j
, P
j
}. Moreover, let 1 q
∞ and let q
′
be the conjugate index of q
1 q
+
1 q
′
= 1 . Set Z
−α q
= Z
α q
′
′
, for 0 α 1, then one has:
T
HEOREM
2. Under the same assumptions of Theorem 1 and with the notations of this Section, we have
Z
−α q
=
f ∈ Z
′
: X
j ≥ j
b
−αq j
k e Q
j
f k
q V
′
+∞
, with
k f k
Z
− α
q
≍ e
P
j
f
V
′
+
X
j ≥ j
b
−αq j
k e Q
j
v k
q V
′
1q
, ∀ f ∈ Z
−α q
.
2.3. Biorthogonal decomposition in
