It is instructive to consider the approximation of signal
xt∈
ℒ
2
x
t
ℒ
2
at coarseness-level-kk of
the Haar system. For the Haar case, projection of
xt∈
ℒ
2
x
t
ℒ
2
onto
V
k
V
k
is accomplished using the basis coefficients

c
k
,
n
=∫−∞∞
φ
k
,
n
txtdt=∫n2k(n+1)2k2−k2xtdt
c
k
,
n
t
φ
k
,
n
t
x
t
t
n
2
k
n
1
2
k
2
k
2
x
t

(1)
giving the approximation

x
k
t=∑n=−∞∞
c
k
,
n
φ
k
,
n
t=∑n=−∞∞∫n2k(n+1)2k2−k2xtdt
φ
k
,
n
t=∑n=−∞∞12k∫n2k(n+1)2kxtdt(2k2
φ
k
,
n
t)
x
k
t
n
c
k
,
n
φ
k
,
n
t
n
t
n
2
k
n
1
2
k
2
k
2
x
t
φ
k
,
n
t
n
1
2
k
t
n
2
k
n
1
2
k
x
t
2
k
2
φ
k
,
n
t

(2)
where

12k∫n2k(n+1)2kxtdt=average value of x(t) in interval
1
2
k
t
n
2
k
n
1
2
k
x
t
average value of x(t) in interval
2k2
φ
k
,
n
t=height=1
k
2
k
2
φ
k
,
n
t
height
1
This corresponds to taking the average value of the signal in
each interval of width

2k
2
k
and approximating the function by a constant over
that interval (see

Figure 1).