150 L. Levaggi – A. Tabacco
and e
V
B +
= span e
ϕ
0k
: S ≤ k ≤ e
L − 1
: = span
e 8
. 87
The systems 8
and e
8 can be biorthogonalized if the matrix
X = hϕ
0k
, e
ϕ
0l
i, with k, l
= S, . . . , ˜L − 1 is non-singular. For example, this condition is verified in the B-spline case see Proposition 12 and also [16].
As before, we set V
j +
: = T
j 0
V
+
and e
V
j +
: = e
T
j 0
e V
+
. Let us define
P : L
p +
−→ V
+
as P
v =
X
k≥S
˘v
0k
ϕ
0k
, ∀v ∈ L
p +
. 88
For any level j 0, let us set P
j
= T
j
◦ P
◦ T
−1 j
. 89
Similar definitions hold for the dual operators e
P
j
. These sequences of operators satisfy the requirements 7, 8 and 9. Following the same construction as in Sections 3 and 5, we build
a multiresolution analysis that will be used to characterize the Besov spaces B
s pq,00
see 3. For the scaling spaces the situation is basically the same, in fact we have only dropped some
functions on the boundary. Many results hold in this context; for example, since
V
+
is a subspace of V
+
, one has
kvk
L
p +
≍
X
k≥S
| ˘v
0k
|
p
1 p
, for any v
∈ V
+
see Proposition 5. We note that while the number of boundary wavelets does not change, they are defined in a
different way since their definition depends on the projector P
. In fact, one has ψ
0,m
∗
−k
: = ϕ
1,k−2k+1 [0,+∞
− P
ϕ
1,k−2k+1 [0,+∞
= ψ
0,m
∗
−k
+
S−1
X
l=0
D ϕ
1,k−2k+1
, e
ϕ
0l
E ϕ
0l
for k = 1, . . . , m
∗
.
7.1. Bernstein and Jackson inequalities
In order to use the characterization Theorems 1 and 2, we will prove Jackson- and Bernstein-type inequalities for Besov spaces B
s pq,0
+
or Sobolev spaces W
s, p +
. Let us observe that the Bernstein inequality follows from 53, because
V
+
⊂ V
+
and B
s pq,0
+
⊂ B
s pq
+
with the same semi-norms. It only remains to prove a Jackson-type inequality, that cannot be deduced directly from 55 because it depends on the projectors 88.
Wavelets on the interval 151
P
ROPOSITION
11. For each s such that 0 S ≤ s ≤ s
L, we have kv −
P
j
v k
L
p +
2
− j s
|v|
B
s pq
+
, ∀v ∈ B
s pq,0
+
, ∀ j ∈
✁
. Proof. We can proceed as in Proposition 7, the only difference being on the first interval I
= [0, 1] and on the space of polynomials to be considered. Indeed, we consider the subspace
✁
L−1
of
✁
L−1
, i.e., the space of polynomials which are zero at zero with all their derivatives of order less than S. Then, inequality 57 holds for every v
∈ B
s pq,0
+
straightforward modifications of the proof of Theorem 4.2, p. 183, in [19].
Finally, we state a characterization theorem for Besov spaces B
s pq,00
+
based on the biorthogonal multilevel decomposition
V
j +
, e
V
j +
as described before. This result immediately follows from Theorem 1.
T
HEOREM
4. Let ϕ ∈ B
s pq
+
for some 0 s L, 1 p, q
∞. For all 0 s s ,
we have B
s pq,00
+
=
v ∈ L
p +
: X
j ≥0
2
sq j
X
k≥S
| ˆv
j k
|
p
q p
∞
90 and
|v|
B
s pq
+
≍
X
j ≥0
2
sq j
X
k≥S
| ˆv
j k
|
p
q p
1 p
, ∀v ∈ B
s pq,00
+
. 91
R
EMARK
5. Since, for any s ∈ , 1 p, q ∞,
1 p
+
1 p
′
=
1 q
+
1 q
′
= 1, B
s pq,00
+ ′
= B
−s p
′
q
′
+
, see [23] p. 235 the extension of the previous theorem to the dual spaces negative s gives the
same result of Theorem 3. R
EMARK
6. As usual, if p = q = 2, s can assume the value 0. In this particular case, if
≤ s L, we get
H
s 00
+
= v
∈ L
2 +
: X
j ≥0
X
k≥S
2
2s j
| ˆv
j k
|
2
+∞ =
H
s +
if s
−
1 2
∈
✁
, H
s 00
+
if s
−
1 2
∈
✁
, and
kvk
H
s +
≍
X
j ≥0
X
k≥S
1 + 2
2s j
| ˆv
j k
|
2
12
, ∀v ∈ H
s 00
+
. 92
152 L. Levaggi – A. Tabacco
R
EMARK
7. In Theorem 4, we have described how to characterize the family of Besov spaces B
s pq,00
+
using a scaling fuction ϕ ∈ B
s pq
+
and removing S = [s
] boundary scaling functions. Of course, one can remove only S S boundary scaling functions and pro-
ceed as before to construct a multiresolution analysis. In this case we characterize B
s pq,00
+
with 0 s ≤ S and B
S pq,00
+
∩ B
s pq
+
for every S s s .
8. Biorthogonal decomposition of the unit interval
