Về một lớp biểu diễn toàn phương thời gian - tần số dựa trên phép biến đổi Fourier thời gian ngắn
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Z
d
+
:
R :
R
d
:
Rez : z
Imz : z
z : z
|z| : z
C
k
(Ω) : {u : Ω → C|u }.
C
∞
0
(Ω) : =
∞
∩
k=0
C
k
0
(Ω), C
k
0
(Ω) C
k
(Ω)
C
0
(R
d
) : .
L
p
: L
p
f
L
p
, f
p
: L
p
.
D
α
f : α f D
α
f(x) = D
α
1
1
D
α
2
2
D
α
n
n
f(x)
D (Ω) :
D
(Ω) :
S(R
d
) :
S
(R
d
) :
STF T :
d
L
2
f
L
2
|f(x)|
2
f(w)
|
f(w)|
2
w f
L
2
=
f
L
2
f(x)
f(w) L
2
f(x) supp f
f(x)
f(w)
2d x ∈ R
d
w ∈ R
d
τ ∈ [0, 1] τ W ig
τ
(f, g)
f, g ∈ S
(R
d
)
[0,1]
W ig
τ
(f, g)(x, w)dτ
•
•
• Q
φ,ψ
• Q
φ,ψ
• Q
φ,ψ
Q
φ,ψ
Q
φ,ψ
•
•
•
Q
φ,ψ
•
•
Ω R
d
D (Ω) ϕ ∈ C
∞
0
(Ω)
{ϕ
j
}
∞
j=1
C
∞
0
(Ω)
ϕ ∈ C
∞
0
(Ω)
K ⊂ Ω supp ϕ
j
⊂ K, j = 1, 2, ,
lim
j→∞
sup
x∈Ω
|D
α
ϕ
j
(x) − D
α
ϕ (x)| = 0, ∀α ∈ Z
d
+
.
ϕ = D lim
j→∞
ϕ
j
.
D(R
d
) ϕ ∈ C
∞
0
(R
d
)
{ϕ
j
}
∞
j=1
⊂ C
∞
0
(R
d
)
ϕ ∈ C
∞
0
(R
d
)
K ⊂ R
d
supp ϕ
j
⊂ K, ∀j = 1, 2, ,
lim
j→∞
sup
x∈R
d
|D
α
ϕ
j
(x) − D
α
ϕ(x)| = 0, ∀α ∈ Z
d
+
D − lim
j→∞
ϕ
j
= ϕ
D (Ω)
S(R
d
)
S(R
d
) =
ϕ ∈ C
∞
(R
d
)
sup
x∈R
d
x
α
D
β
ϕ(x)
< +∞, ∀α, β ∈ Z
d
+
{ϕ
k
}
∞
k=1
⊂ S
R
d
ϕ ∈ S
R
d
S
R
d
lim
k→∞
sup
x∈R
d
x
α
D
β
ϕ
k
(x) − x
α
D
β
ϕ(x)
= 0, ∀α, β ∈ Z
d
+
.
S lim
k→∞
ϕ
k
= ϕ
ϕ ∈ C
∞
(R
d
)
m ∈ Z
+
, β ∈ Z
d
+
(1 + |x|
2
)
m
D
β
ϕ(x)
≤ C
m,β
, ∀x ∈ R
d
,
m ∈ Z
+
(1 + |x|
2
)
m
|β|≤m
D
β
ϕ(x)
≤ C
m
, ∀x ∈ R
d
.
λ, µ ∈ C; ϕ
k
, ψ
k
, ϕ, ψ ∈ S(R
d
), k = 1, 2, S lim
k→∞
ϕ
k
=
ϕ, S lim
k→∞
ψ
k
= ψ S lim
k→∞
(λϕ
k
+ µψ
k
) = λϕ + µψ
C
∞
0
(R
d
) S(R
d
)
a(·) ∈ C
∞
(R
d
) α ∈ Z
d
+
m = m(α)
c = c(α) |D
α
a (x)| < c (1 + |x|)
m
ϕ aϕ S(R
d
) S(R
d
)
S
R
d
f Ω f
D (Ω)
Ω D
(Ω) f ∈ D
(Ω)
ϕ ∈ D (Ω) f, ϕ f, g
f, ϕ = g, ϕ, ∀ϕ ∈ D (Ω) .
f ∈ D
(Ω) α =
(α
1
, α
2
, , α
d
) ∈ Z
d
+
α f Ω
D
α
f D (Ω) C
D
α
f : ϕ → (−1)
|α|
f, D
α
ϕ, ϕ ∈ D (Ω) .
S
(R
d
)
D
(R
d
)
D(R
d
) f ∈ D
(R
d
)
ϕ ∈ D(R
d
) f, ϕ
{f
k
}
∞
k=1
f D
(R
d
) k
lim
k→∞
f
k
, ϕ = f, ϕ, ∀ϕ ∈ D(R
d
).
D
lim
k→∞
f
k
= f.
f ∈ D
(R
d
) f
m ∈ N C
|f, ϕ| C sup
x∈R
d
(1 + |x|
2
)
m
|α|m
|D
α
ϕ(x)|, ∀ϕ ∈ D(R
d
)
f S
(R
d
)
f
k
, f ∈ S
(R
d
), k = 1, 2, {f
k
}
∞
k=1
S
(R
d
) f ∈ S
(R
d
)
m C
|f
k
, ϕ| C sup
x∈R
d
(1+|x|
2
)
m
|α|m
|D
α
ϕ(x)|, ∀ϕ ∈ C
∞
0
(R
d
), k = 1, 2, ,
{f
k
}
∞
k=1
D
(R
d
) f
S
lim
k→∞
f
k
= f
S
R
d
f R
d
f
Z
d
f(x) = f(x + k) k ∈ Z
d
Z
d
[0; 1]
d
[0; 1]
d
R
d
Z
d
d− T
d
:= R
d
/Z
d
e
2πin.x
, n ∈ Z
d
T
d
[0; 1]
d
L
2
T
d
f ∈ L
2
T
d
f (n) =
[0,1]
d
f (x) e
−2πin.x
dx, n ∈ Z
d
n f
f =
n∈Z
d
f (n) e
2πin.x
[0,1]
d
|f (x)|
2
dx = f
2
L
2
(T
d
)
=
n∈Z
d
f (n)
2
.
T
d
n∈Z
d
f (n)
< ∞ f (x) , ∀x ∈ R
d
f(x)
A
T
d
f
A
=
n∈Z
d
f (n)
A
T
d
f ∈ D
T
d
= (C
∞
)
x.w =
d
i=1
x
i
w
i
R
d
x
2
= x.x
|x| =
√
x.x
f ∈ L
1
R
d
f
f (w) =
R
d
e
−2πix.w
f (x) dx, w ∈ R
d
.
Ff
f 2π
w
f(w)
w w
|
f (w)|
2
f
2
2
f
−2
2
I
|
f (w)|
2
dw
f I ⊆ R
d
L
1
f
∞
≤ f
1
.
f ∈ L
1
R
d
f
R
d
lim
|w|→∞
f (w)
= 0.
C
0
R
d
F : L
1
R
d
→ C
0
R
d
.
f ∈ L
1
∩ L
2
R
d
f
2
=
f
2
.
F L
2
(R
d
)
f, g =
f, g
, ∀f, g ∈ L
2
R
d
.
f ∈ L
1
R
d
f ∈ L
1
R
d
f (x) =
R
d
f (w) e
2πix.w
dw x ∈ R
d
.
F
−1
= F.
f (x) = f (−x)
f
e
2πix.w
f
L
1
(R
d
)
S(R
d
)
ϕ ∈ S(R
d
) ϕ Fϕ
Fϕ (ξ) = (2π)
−
d
2
R
d
e
−ix.ξ
ϕ (x) dx.
S(R
d
)
ϕ F
−1
ϕ
F
−1
ϕ (ξ) = (2π)
−
d
2
R
d
e
ix.ξ
ϕ (x) dx.
e
−ix.ξ
=
e
ix.ξ
= 1, ϕ ∈ L
1
R
d
Fϕ
F
−1
ϕ R
d
Fϕ(ξ) =
F
−1
ϕ(−ξ), F
−1
ϕ (ξ) = Fϕ (ξ) f ∈ L
1
R
d
Ff, F
−1
f ∈ C
0
R
d
u ∈ S
R
d
u Fu u
Fu, ϕ = u (ϕ) = u ( ϕ) , ∀ϕ ∈ S
R
d
.
F
−1
(u) = F [u (−x)] , u ∈ S
R
d
.
α = (α
1
, , α
d
) ∈ Z
d
+
|α| =
d
j=1
α
j
, w
α
=
d
j=1
w
α
j
j
,
D
α
=
∂
α
1
∂x
α
1
1
∂
α
d
∂x
α
d
d
α X
α
f (x) = x
α
f (x)
(D
α
f)
∧
(w) = (2πiw)
α
f (w)
((−2πix)
α
f)
∧
(w) = D
α
f (w)
FD
α
= (2πi)
|α|
X
α
F FX
α
=
i
2π
|α|
D
α
F
f ∈ C
∞
R
d
D
α
f (w) =
R
d
f (x) D
α
e
−2πix.w
dx =
R
d
(−2πix)
α
f (x) e
−2πix.w
dx.
x, w ∈ R
d
x
T
x
f (t) = f (t − x) .
M
w
f (t) = e
2πiw.t
f (t) .
f, g ∈ L
1
R
d
f ∗ g
(f ∗ g) (x) =
R
d
f (y) g (x −y) dy.
f
∗
(x) = f (−x)
f (x) = f (−x)
f ∗ g f
1
g
1
(f ∗ g) =
f.g
T
x
M
w
f(t) = e
−2πix.w
M
w
T
x
f (t)
(T
x
f) = M
−x
f
(M
w
f) = T
w
f
T
x
M
w
f = M
−x
T
w
f = e
−2πix.w
T
w
M
−x
f.
f
∗
=
f
f = I
f
(f ∗ g) (x) = f, T
x
g
∗
R
f ∈ L
1
R
d
α > 0
R
d
f (x) dx =
[0,α]
d
k∈Z
d
f (x + αk)
dx.
αk +[0, α]
d
0
R
d
R
d
f (x) dx =
k∈Z
d
αk+[0,α]
d
f (x) dx =
[0,α]
d
k∈Z
d
f (x + αk) dx.
f ∈ L
1
R
d
ε > 0 C > 0
|f (x)| ≤ C (1 + |x|)
−d−ε
f (w)
≤ C (1 + |w|)
−d−ε
n∈Z
d
f (x + n) =
n∈Z
d
f (n) e
2πin.x
.
∀x ∈ R
d
x ∈ R
d
ϕ(x) =
n∈Z
d
f(x + n) Z
d
f ∈ L
1
R
d
ϕ ∈ L
1
(T
d
)
ϕ(n) =
[0,1]
d
ϕ(x)e
−2πin.x
dx
=
[0,1]
d
k∈Z
d
f(x + k)e
−2πin.(x−k)
dx
=
R
d
f(x)e
−2πin.x
dx =
f(n).
n∈Z
d
f(n)
< ∞, ϕ
ϕ(x) =
k∈Z
d
f(n)e
2πin.x
ϕ
a
(x) = e
−
πx
2
a
a > 0 R
d
a > 0
ϕ
a
(w) = a
d/2
ϕ
1/a
(w) .
a = 1,
e
−πx
2
∧
= e
−πw
2
a > 0 x, u, w, η ∈ R
d
T
x
M
w
ϕ
a
, T
u
M
η
ϕ
a
=
a
2
d/2
e
πi(u−x)(η+w)
ϕ
2a
(u − x) ϕ
2
a
(η −w) .
ϕ
a
, M
w
T
x
ϕ
a
=
R
d
e
−πt
2
/a
e
−π(t−x)
2
/a
e
−2πiw.t
dt
= e
−πx
2
/(2a)
R
d
e
−2π(t−x/2)
2
/a
e
−2πiw.t
dt
= ϕ
2a
(x)
T
x
2
ϕ
a
2
∧
(w)
= e
−πix.w
a
2
d
2
ϕ
2a
(x)ϕ
2
a
(w).
M
−w
T
u−x
M
η
= e
−2πiη.(u−x)
M
η−w
T
u−x
T
x
M
w
ϕ
a
, T
u
M
η
ϕ
a
= ϕ
a
, M
−w
T
u−x
M
η
ϕ
a
= e
2πiη.(u−x)
ϕ
a
, M
η−w
T
u−x
ϕ
a
=
a
2
d
2
e
2πiη.(u−x)
e
−πi(u−x).(η−w)
ϕ
2a
(u − x)ϕ
2
a
(η −w)
=
a
2
d
2
e
πi(u−x).(η+w)
ϕ
2a
(u − x)ϕ
2
a
(η −w).
a > 0
T
x
M
w
ϕ
a
: x, w ∈ R
d
L
2
R
d
X
ϕ
1
X = span
T
x
M
w
ϕ
1
: x, w ∈ R
d
(T
x
M
w
ϕ
1
)
∧
= M
−x
T
w
ϕ
1
= e
−2πi x.w
T
w
M
−x
ϕ
1
,
X F
L
2
R
d
T
x
M
w
ϕ
1
, T
u
M
η
ϕ
1
= F (T
x
M
w
ϕ
1
) , F (T
u
M
η
ϕ
1
),
X F X X
L
2
R
d
T
x
M
w
ϕ
1
, T
u
M
η
ϕ
1
= 2
−d/2
e
πi(u−x).(η+w)
ϕ
2
(u − x) ϕ
2
(η −w) .
F (T
x
M
w
ϕ
1
) , F (T
u
M
η
ϕ
1
)
= e
−2πi(x.w−u.η)
T
w
M
−x
ϕ
1
, T
η
M
−u
ϕ
1
= 2
−d/2
e
−2πi(x.w−uη)
e
πi(η−w).(−x−u)
ϕ
2
(η −w) ϕ
2
(−u + x)
= T
x
M
w
ϕ
1
, T
u
M
η
ϕ
1
.
f =
n
k=1
c
k
T
x
k
M
w
k
ϕ
1
X
f
2
2
=
n
k,l=1
c
k
c
l
T
x
k
M
w
k
ϕ
1
, T
x
l
M
w
l
ϕ
1
=
n
k,l=1
c
k
c
l
F(T
x
k
M
w
k
ϕ
1
), F(T
x
l
M
w
l
ϕ
1
)
=
f,
f
=
f
2
2
.
f ∈ X
g = 0
f g
V
g
f(x, w) =
R
d
f(t)g(t − x)e
−2πit.w
dt, ∀x, w ∈ R
d
.
g V
g
f(x, ·)
f
x x x
V
g
f(x, w)
w x V
g
f(x, ·)
x
d = 1 R
2d
R
2d
f
g g V
g
f
R
d
R
2d
V
g
f f → V
g
f
g
f, g ∈ L
2
R
d
V
g
f R
2d
V
g
f (x, w) = (f · T
x
g)
∧
(w)
= f, M
w
T
x
g
=
f, T
w
M
−x
g
= e
−2πix.w
f · T
w
g
∧
(−x)
= e
−2πix.w
V
g
f (w, −x)
= e
−2πix.w
(f ∗ M
w
g
∗
) (x)
=
f ∗ M
−x
g
∗
(w)
= e
−πix.w
R
d
f
t +
x
2
g
t −
x
2
e
−2πit.w
dt.
g
(f, g) →
V
g
f f ⊗g f ⊗ g(x, t) = f(x)g(t) T
a
T
a
F (x, t) = F (t, t − x) F
2
F
2
F (x, w) =
R
d
F (x, t)e
−2πit.w
dt F
R
2d
f, g ∈ L
2
(R
d
)
V
g
f = F
2
T
a
(f ⊗
g).
V
g
f
V
g
(T
u
M
η
f)(x, w) = e
−2πiu.w
V
g
f(x − u, w − η),
x, u, w, η ∈ R
d
|V
g
(T
u
M
η
f)(x, w)| = |V
g
f(x − u, w − η)|.
f
1
, f
2
, g
1
, g
2
∈ L
2
R
d
V
g
j
f
j
∈ L
2
R
2d
j = 1, 2
V
g
1
f
1
, V
g
2
f
2
L
2
(R
2d
)
= f
1
, f
2
g
1
, g
2
.
f, g ∈ L
2
(R)
V
g
f
2
= f
2
g
2
.
g
2
= 1
f
2
= V
g
f
2
, f ∈ L
2
R
d
.
L
2
R
d
L
2
R
2d
g, γ ∈
L
2
R
d
g, γ = 0 f ∈ L
2
R
d
f =
1
γ, g
R
2d
V
g
f (x, w) M
w
T
x
γdwdx.
V
g
f ∈ L
2
R
d
f =
1
γ, g
R
2d
V
g
f (x, w) M
w
T
x
γdxdw
L
2
R
d
f, h
=
1
γ, g
R
2d
V
g
f (x, w)
h, M
w
T
x
γ
dxdw
=
1
γ, g
V
g
f, V
γ
h = f, h.
f = f
g, γ ∈ L
2
(R
d
) K
n
⊆ R
2d
n ≥ 1
n≥1
K
n
= R
2d
K
n
⊆ int K
n+1
, n = 1, 2, f
n
f
n
=
1
γ, g
K
n
V
g
f (x, w) M
w
T
x
γdxdw.
lim
n→∞
f − f
n
2
= 0.
f
2
= g
2
= 1 U ⊆ R
2d
ε 0
U
|V
g
f (x, w)|
2
dxdw 1 − ε.
|U| 1 − ε
|V
g
f (x, w)| = |f, M
w
T
x
g| f
2
g
2
(x, w) ∈ R
2d
.
1 − ε
U
|V
g
f (x, w)|
2
dxdw V
g
f
2
∞
|U| ≤ |U|.
f, g ∈ L
2
R
d
2 p < ∞
R
2d
|V
g
f (x, w)|
p
dxdw
2
p
d
(f
2
g
2
)
p
.
f ∈ L
2
R
d
Af(x, w) =
R
d
f
t +
x
2
f
t −
x
2
e
−2πitw
dt
= e
πix.w
V
f
(x, w) .
f g ∈ L
2
R
d
A (f, g) (x, w) =
R
d
f
t +
x
2
g
t −
x
2
e
−2πitw
dt
= e
2πit.w
V
g
f (x, w) .
(Af)
∗
(x, w) = Af (−x, −w)) = Af (x, w) .
V
f
f
2
= f
2
f
2
= f
2
2
|Af (x, w)|
2
=
|V
f
(x, w)|
2
R
2d
|Af (x, w)|
2
dxdw =
R
2d
|V
f
(x, w)|
2
dxdw = V
f
f (x, w)
2
= f
4
2
.
f
2
= 1 U ⊂ R
2d
ε > 0
U
|Af (x, w)|
2
dxdw 1 − ε,
|U| 1 − ε.
