Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be the right kind of function spaces for time-frequency analysis. Feichtinger's algebra, while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis.
Modulation spaces are defined as follows. For
1
≤
p
,
q
≤
∞
, a non-negative function
m
(
x
,
ω
)
on
R
2
d
and a test function
g
∈
S
(
R
d
)
, the modulation space
M
m
p
,
q
(
R
d
)
is defined by
M
m
p
,
q
(
R
d
)
=
{
f
∈
S
′
(
R
d
)
:
(
∫
R
d
(
∫
R
d
|
V
g
f
(
x
,
ω
)
|
p
m
(
x
,
ω
)
p
d
x
)
q
/
p
d
ω
)
1
/
q
<
∞
}
.
In the above equation,
V
g
f
denotes the short-time Fourier transform of
f
with respect to
g
evaluated at
(
x
,
ω
)
, namely
V
g
f
(
x
,
ω
)
=
∫
R
d
f
(
t
)
g
(
t
−
x
)
¯
e
−
2
π
i
t
⋅
ω
d
t
=
F
ξ
−
1
(
g
^
(
ξ
)
¯
f
^
(
ξ
+
ω
)
)
(
x
)
.
In other words,
f
∈
M
m
p
,
q
(
R
d
)
is equivalent to
V
g
f
∈
L
m
p
,
q
(
R
2
d
)
. The space
M
m
p
,
q
(
R
d
)
is the same, independent of the test function
g
∈
S
(
R
d
)
chosen. The canonical choice is a Gaussian.
We also have a Besov-type definition of modulation spaces as follows.
M
p
,
q
s
(
R
d
)
=
{
f
∈
S
′
(
R
d
)
:
(
∑
k
∈
Z
d
⟨
k
⟩
s
q
∥
ψ
k
(
D
)
f
∥
p
q
)
1
/
q
<
∞
}
,
⟨
x
⟩
:=
|
x
|
+
1
,
where
{
ψ
k
}
is a suitable unity partition. If
m
(
x
,
ω
)
=
⟨
ω
⟩
s
, then
M
p
,
q
s
=
M
m
p
,
q
.
For
p
=
q
=
1
and
m
(
x
,
ω
)
=
1
, the modulation space
M
m
1
,
1
(
R
d
)
=
M
1
(
R
d
)
is known by the name Feichtinger's algebra and often denoted by
S
0
for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators.
M
1
(
R
d
)
is a Banach space embedded in
L
1
(
R
d
)
∩
C
0
(
R
d
)
, and is invariant under the Fourier transform. It is for these and more properties that
M
1
(
R
d
)
is a natural choice of test function space for time-frequency analysis. Fourier transform
F
is an automorphism on
M
1
,
1
.
