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
The abstract setting can be applied to construct a biorthogonal system of compactly supported wavelets on the real line and a biorthogonal decomposition of L
p
1 p ∞. We will be
very brief and we refer to [5, 6, 11, 17] for proofs and more details. We suppose to have a couple of dual compactly supported scaling functions ϕ
∈ L
p
and e
ϕ ∈ L
p
′
1 p
+
1 p
′
= 1 satisfying the following conditions. There exists a couple of finite
real filters h = {h
n
}
n
1
n=n
, ˜ h
= ˜h
n ˜
n
1
n= ˜ n
with n ,
˜n ≤ 0 and n
1
, ˜n
1
≥ 0, so that ϕ and e ϕ
satisfy the refinement equations: ϕ
x =
√ 2
n
1
X
n=n
h
n
ϕ 2x
− n , e
ϕ x
= √
2
˜ n
1
X
n= ˜ n
˜h
n
e ϕ
2x − n ,
25 and
supp ϕ = [n
, n
1
] , supp
e ϕ
= ˜n
, ˜n
1
. From now on, we will only describe the primal setting, the parallel
˜ construction following by analogy; it will be understood that there p has to be replaced by the conjugate index p
′
. Setting, as usual, for j, k
∈
✂
, ϕ
j k
x = 2
j p
ϕ 2
j
x − k, we have the biorthogonality
relations hϕ
j k
, e
ϕ
j k
′
i = Z
ϕ
j k
x e
ϕ
j k
′
x d x = δ
kk
′
, ∀ j, k, k
′
∈
✂
.
130 L. Levaggi – A. Tabacco
Thus 8
j
= {ϕ
j k
: k ∈
✂
} are uniformly p-stable bases for the spaces V
j
= V
j
= span
L
p
{ϕ
j k
: k ∈
✂
} =
X
k∈
α
k
ϕ
j k
: {α
k
}
k∈
∈ ℓ
p
,
and, for any v ∈ V
j
, we can write v
= X
k∈
α
k
ϕ
j k
= X
k∈
˘v
j k
ϕ
j k
with ˘v
j k
= hv, e ϕ
j k
i and
kvk
L
p
≍
X
k∈
| ˘v
j k
|
p
1 p
. 26
We have V
j
⊂ V
j +1
, \
j ∈
V
j
= {0} , [
j ∈
V
j
= L
p
. We suppose there exists an integer L
≥ 1 so that, locally, the polynomials of degree up to L − 1 we will indicate this set
✁
L−1
are contained in V
j
. It is not difficult to show that L must satisfy the relation
L ≤ n
1
− n 27
see [6], equation 3.2. We set
ψ x
= √
2
1− e
n
X
n=1− e
n
1
g
n
ϕ 2x
− n , with g
n
= −1
n
e h
1−n
, and ψ
j k
x = 2
j p
ψ 2
j
x − k, ∀ j, k ∈
✂
. Then hψ
j k
, e ψ
j
′
k
′
i = δ
j j
′
δ
kk
′
, ∀ j, j
′
, k, k
′
∈
✂
. The wavelet spaces
W
j
=
X
k∈
α
k
ψ
j k
: {α
k
}
k∈
∈ ℓ
p
,
∀ j ∈
✂
, satisfy L
p
= ⊕
j ∈
W
j
. For any v ∈ L
p
, this implies the expansion v
= X
j,k∈
ˆv
j k
ψ
j k
, with
ˆv
j k
= hv, ˜ ψ
j k
i ; 28
in addition, if p = 2,
kvk
L
2
≍
X
j,k∈
| ˆv
j k
|
2
12
, ∀v ∈ L
2
. 29
Next, let us recall that if ϕ ∈ B
s pq
s 0, 1 p, q
+∞ then the Bernstein and Jackson inequalities hold for every space Z
= B
s pq
with 0 ≤ s mins
, L. Indeed, we have
|v|
B
s pq
2
j s
kvk
L
p
, ∀v ∈ V
j
, ∀ j ∈
✂
30
Wavelets on the interval 131
and kv − P
j
v k
L
p
2
− j s
|v|
B
s pq
, ∀v ∈ B
s pq
, ∀ j ∈
✂
. 31
Thus, taking into account 1, we can apply the characterization Theorems 1 and 2 to the Besov space Z .
3. Scaling function spaces for the half-line
