The *difference* in detail between
V
k
V
k
and
V
k
−
1
V
k
−
1
will be described using
W
k
W
k
, the orthogonal complement of
V
k
V
k
in
V
k
−
1
V
k
−
1
:

V
k
−
1
=
V
k
⊕
W
k
V
k
−
1
V
k
W
k

(1)
At times it will be convenient to write

W
k
=
V
k
⊥
W
k
V
k
⊥
. This concept is illustrated in the set-theoretic
diagram,

Figure 1.

Suppose now that, for each
k∈Z
k
, we construct an orthonormal basis for
W
k
W
k
and denote it by
ψ
k
,
n
t
n∈Z
ψ
k
,
n
t
n
. It turns out that, because every
V
k
V
k
has a basis constructed from shifts and stretches of a mother
scaling function (i.e.,
φ
k
,
n
t=2−k2φ2−kt−n
φ
k
,
n
t
2
k
2
φ
2
k
t
n
, every
W
k
W
k
has a basis that can be constructed from shifts and stretches
of a "mother wavelet"
ψt∈
ℒ
2
ψ
t
ℒ
2
:
ψ
k
,
n
t=2−k2ψ2−kt−n
.
ψ
k
,
n
t
2
k
2
ψ
2
k
t
n
.
The Haar system will soon provide us with a concrete example
.

Let's focus, for the moment, on the specific case
k=1
k
1
. Since
W1⊂V0
W1
V0
, there must exist
gn
n∈Z
g
n
n
such that:

ψ
1
,
0
t=∑n=−∞∞gn
φ
0
,
n
t
ψ
1
,
0
t
n
g
n
φ
0
,
n
t

(2)
⇔
12ψ12t=∑n=−∞∞gnφt−n
⇔
1
2
ψ
1
2
t
n
g
n
φ
t
n
ψt=2∑n=−∞∞gnφ2t−n
ψ
t
2
n
g
n
φ
2
t
n

(3)
To be a valid scaling-function/wavelet pair,

φt
φ
t
and

ψt
ψ
t
must obey the wavelet scaling equation for some
coefficient set

gn
g
n
.