Fourier series is a useful orthonormal representation on
L
2
(
[
0
,
T
]
)
L
2
(
[
0
,
T
]
)
especiallly for inputs into LTI systems. However, it is ill suited for
some applications, i.e. image processing (recall Gibb's phenomena).
Wavelets, discovered in the last 15 years, are another
kind of basis for
L
2
(
[
0
,
T
]
)
L
2
(
[
0
,
T
]
)
and have many nice properties.
Fourier series 
c
n
c
n
give frequency information. Basis functions last the entire
interval.
Wavelets  basis functions give frequency info but are
local in time.
In Fourier basis, the basis functions are harmonic
multiples of
ei
ω
0
t
ω
0
t
In Haar wavelet basis, the basis functions are scaled and
translated versions of a "mother wavelet"
ψt
ψ
t
.
Basis functions
ψ
j
,
k
t
ψ
j
,
k
t
are indexed by a scale j and a
shift k.
Let
∀0≤t<T:φt=1
0
t
T
φ
t
1
Then
φt
2j2ψ2jt−k
φt2j2ψ2jt−k
j∈ℤ∧(k=
0
,
1
,
2
,
…
,
2
j

1
)
φ
t
2
j
2
ψ
2
j
t
k
j
ℤ
k
0
,
1
,
2
,
…
,
2
j

1
φ
t
2
j
2
ψ
2
j
t
k
ψt={1 if 0≤t<T21 if 0≤T2<T
ψ
t
1
0
t
T
2
1
0
T
2
T
(1)
Let
ψ
j
,
k
t=2j2ψ2jt−k
ψ
j
,
k
t
2
j
2
ψ
2
j
t
k
Larger jj → "skinnier" basis
function,
j=012…
j
0
1
2
…
,
2j
2
j
shifts at each scale:
k=
0
,
1
,
…
,
2
j

1
k
0
,
1
,
…
,
2
j

1
Check: each
ψ
j
,
k
t
ψ
j
,
k
t
has unit energy
(∫
ψ
j
,
k
2tdt=1)⇒(
∥
ψ
j
,
k
(
t
)
∥
2
=1)
t
ψ
j
,
k
t
2
1
∥
ψ
j
,
k
(
t
)
∥
2
1
(2)
Any two basis functions are orthogonal.
Also,
ψ
j
,
k
φ
ψ
j
,
k
φ
span
L
2
(
[
0
,
T
]
)
L
2
(
[
0
,
T
]
)
Using what we know about Hilbert spaces: For any
ft∈
L
2
(
[
0
,
T
]
)
f
t
L
2
(
[
0
,
T
]
)
,
we can write
ft=∑j∑k
w
j
,
k
ψ
j
,
k
t+
c
0
φt
f
t
j
j
k
k
w
j
,
k
ψ
j
,
k
t
c
0
φ
t
(3)
w
j
,
k
=∫0Tft
ψ
j
,
k
tdt
w
j
,
k
t
0
T
f
t
ψ
j
,
k
t
(4)
c
0
=∫0Tftφtdt
c
0
t
0
T
f
t
φ
t
(5)
the
w
j
,
k
w
j
,
k
are real
The Haar transform is
super useful especially in
image compression