The coefficients
hn
h
n
were originally introduced at describe
φ
1
,
0
t
φ
1
,
0
t
in terms of the basis for
V
0
V
0
:
φ
1
,
0
t=∑nhn
φ
0
,
n
t
.
φ
1
,
0
t
n
h
n
φ
0
,
n
t
.
From the previous equation we find that

φ
0
,
m
t ·
φ
1
,
0
t=
φ
0
,
m
t · ∑nhn
φ
0
,
n
t=∑nhn
φ
0
,
m
t ·
φ
0
,
n
t=hm
φ
0
,
m
t
φ
1
,
0
t
φ
0
,
m
t
n
h
n
φ
0
,
n
t
n
h
n
φ
0
,
m
t
φ
0
,
n
t
h
m

(1)
where

δn−m=
φ
0
,
m
t ·
φ
0
,
n
t
δ
n
m
φ
0
,
m
t
φ
0
,
n
t
, which gives a way to calculate the coefficients

hm
h
m
when we know

φ
k
,
n
t
φ
k
,
n
t
.

In the Haar case

hm=∫−∞∞
φ
0
,
m
t
φ
1
,
0
tdt=∫mm+1
φ
1
,
0
tdt={12 if m∈010 otherwise
h
m
t
φ
0
,
m
t
φ
1
,
0
t
t
m
m
1
φ
1
,
0
t
1
2
m
0
1
0

(2)
since

φ
1
,
0
t=12
φ
1
,
0
t
1
2
in the interval

0
2
0
2
and zero otherwise. Then choosing

P=1
P
1
in

gn=-1nhP−n
g
n
-1
n
h
P
n
, we find that

gn={12 if 0−12 if 10 otherwise
g
n
1
2
0
1
2
1
0
for the Haar system. From the wavelet scaling equation

ψt=2∑ngnφ2t−n=φ2t−φ2t−1
ψ
t
2
n
g
n
φ
2
t
n
φ
2
t
φ
2
t
1
we can see that the Haar mother wavelet and scaling function
look like in

Figure 1:

It is now easy to see, in the Haar case, how integer shifts of
the mother wavelet describe the differences between signals in
V
−
1
V
−
1
and
V
0
V
0
(Figure 2):

We expect this because
V
−
1
=
V
0
⊕
W
0
V
−
1
V
0
W
0
.