Recall that
V
k
=
W
k
+
1
⊕
V
k
+
1
V
k
W
k
+
1
V
k
+
1
and that
V
k
+
1
=
W
k
+
2
⊕
V
k
+
2
V
k
+
1
W
k
+
2
V
k
+
2
. Putting these together and extending the idea
yields

V
k
=
W
k
+
1
⊕
W
k
+
2
⊕
V
k
+
2
=
W
k
+
1
⊕
W
k
+
2
⊕…⊕
W
ℓ
⊕
V
ℓ
=
W
k
+
1
⊕
W
k
+
2
⊕
W
k
+
3
⊕…=
⊕
i
=
k
+
1
∞
W
i
V
k
W
k
+
1
W
k
+
2
V
k
+
2
W
k
+
1
W
k
+
2
…
W
ℓ
V
ℓ
W
k
+
1
W
k
+
2
W
k
+
3
…
⊕
i
=
k
+
1
∞
W
i

(1)
If we take the limit as

k→−∞
k
, we find that

ℒ
2
=
V
−
∞
=
⊕
i
=
−
∞
∞
W
i
ℒ
2
V
−
∞
⊕
i
=
−
∞
∞
W
i

(2)
Moreover,

(W1⊥V1)∧(
W
k
≥
2
⊂
V
1
)⇒(W1⊥
W
k
≥
2
)
⊥
W1
V1
W
k
≥
2
V
1
⊥
W1
W
k
≥
2

(3)
(W2⊥V2)∧(
W
k
≥
3
⊂V2)⇒(W2⊥
W
k
≥
3
)
⊥
W2
V2
W
k
≥
3
V2
⊥
W2
W
k
≥
3

(4)
from which it follows that

Wk⊥
W
j
≠
k
⊥
Wk
W
j
≠
k

(5)
or, in other words, all subspaces

W
k
W
k
are orthogonal to one another. Since the functions

ψ
k
,
n
t
n∈Z
ψ
k
,
n
t
n
form an orthonormal basis for

W
k
W
k
, the results above imply that

ψ
k
,
n
t
n∧k∈Z
constitutes an orthonormal basis for
ℒ
2
ψ
k
,
n
t
n
k
constitutes an orthonormal basis for
ℒ
2

(6)
This implies that, for any

ft∈
ℒ
2
f
t
ℒ
2
, we can write

ft=∑k=−∞∞∑m=−∞∞
d
k
m
ψ
k
,
m
t
f
t
k
m
d
k
m
ψ
k
,
m
t

(7)
d
k
m=
ψ
k
,
m
t · ft
d
k
m
ψ
k
,
m
t
f
t

(8)
This is the key idea behind the orthogonal wavelet system that
we have been developing!