The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. It is infinitely differentiable with infinite support and defined in frequency domain in terms of function
ν
as:
Ψ
(
ω
)
:=
{
1
2
π
sin
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
e
j
ω
/
2
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
4
π
−
1
)
)
e
j
ω
/
2
if
4
π
/
3
<
|
ω
|
<
8
π
/
3
,
0
otherwise
,
where:
ν
(
x
)
:=
{
0
if
x
<
0
,
x
if
0
<
x
<
1
,
1
if
x
>
1.
There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet. For instance, another standard implementation adopts
ν
(
x
)
:=
{
x
4
(
35
−
84
x
+
70
x
2
−
20
x
3
)
if
0
<
x
<
1
,
0
otherwise
.
The Meyer scale function is given by:
Φ
(
ω
)
:=
{
1
2
π
if
|
ω
|
<
2
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
0
otherwise
.
In the time-domain, the waveform of the Meyer mother-wavelet has the shape as shown in the following figure:
In 2015, Victor Vermehren Valenzuela and H. M. de Oliveira gave the explicitly expressions of Meyer wavelet and scale functions:
ϕ
(
t
)
=
{
2
3
+
4
3
π
t
=
0
,
sin
(
2
π
3
t
)
+
4
3
t
cos
(
4
π
3
t
)
π
t
−
16
π
9
t
3
o
t
h
e
r
w
i
s
e
,
and
ψ
(
t
)
=
ψ
1
(
t
)
+
ψ
2
(
t
)
where
ψ
1
(
t
)
=
4
3
π
(
t
−
1
2
)
cos
[
2
π
3
(
t
−
1
2
)
]
−
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
16
9
(
t
−
1
2
)
3
,
and
ψ
2
(
t
)
=
8
3
π
(
t
−
1
2
)
cos
[
8
π
3
(
t
−
1
2
)
]
+
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
64
9
(
t
−
1
2
)
3
.
