An orthonormal wavelet basis is an orthonormal basis of the form

B=
2j2ψ2jt−k
j∈Z∧k∈Z
B
2
j
2
ψ
2
j
t
k
j
k

(1)
The function

ψt
ψ
t
is called the

wavelet.

The problem is how to find a function
ψt
ψ
t
so that the set BB is an
orthonormal set.

The Haar basis
(described by Haar in 1910) is an orthonormal
basis with wavelet
ψt
ψ
t

ψt={1 if 0≤t≤1/2-1 if 1/2≤t≤10 otherwise
ψ
t
1
0
t
12
-1
12
t
1
0

(2)
For the Haar wavelet, it is easy to verify that the set

BB is an orthonormal set (

Figure 1).

Notation:
ψ
j
,
k
t=2j2ψ2jt−k
ψ
j
,
k
t
2
j
2
ψ
2
j
t
k
where jj is an index of
*scale* and kk
is an index of *location*.

If BB is an orthonormal set then
we have the wavelet series.

xt=∑j=−∞∞∑k=−∞∞djk
ψ
j
,
k
t
x
t
j
k
d
j
k
ψ
j
,
k
t

(3)
djk=∫−∞∞xt
ψ
j
,
k
tdt
d
j
k
t
x
t
ψ
j
,
k
t