Let
⋯
⊂
V
−
1
⊂
V
0
⊂
V
1
⊂
⋯
\cdots \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset \cdots
be a multiresolution analysis of
L
2
{L^2}
generated by the
m
m
th order
B
B
-spline
N
m
(
x
)
{N_m}(x)
. In this paper, we exhibit a compactly supported basic wavelet
ψ
m
(
x
)
{\psi _m}(x)
that generates the corresponding orthogonal complementary wavelet subspaces
⋯
,
W
−
1
,
W
0
,
W
1
,
…
\cdots ,{W_{ - 1}},{W_0},{W_1}, \ldots
. Consequently, the two finite sequences that describe the two-scale relations of
N
m
(
x
)
{N_m}(x)
and
ψ
m
(
x
)
{\psi _m}(x)
in terms of
N
m
(
2
x
−
j
)
,
j
∈
Z
{N_m}(2x - j),j \in \mathbb {Z}
, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases
{
N
~
m
(
x
−
j
)
}
\{ {\tilde N_m}(x - j)\}
and
{
ψ
~
m
(
x
−
j
)
}
\{ {\tilde \psi _m}(x - j)\}
, relative to
{
N
m
(
x
−
j
)
}
\{ {N_m}(x - j)\}
and
{
ψ
m
(
x
−
j
)
}
\{ {\psi _m}(x - j)\}
, respectively.