Seien
f
,
g
:
R
→
C
{\displaystyle f,g\colon \mathbb {R} \to \mathbb {C} }
T
{\displaystyle T}
⟨
f
,
g
⟩
:=
1
T
∫
t
0
t
0
+
T
f
(
t
)
¯
g
(
t
)
d
t
{\displaystyle \langle f,g\rangle :={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}{\overline {f(t)}}\,g(t)\,\mathrm {d} t}
Bemerkung: Im Fall einer rein reellen Funktion ist die Konjugation wirkungslos und kann somit entfallen.
Diese Operation ist ein Skalarprodukt auf dem Hilbertraum
L
2
(
t
0
,
t
0
+
T
)
{\displaystyle L^{2}(t_{0},t_{0}+T)}
Manchmal ist die Interpretation von
t
{\displaystyle t}
x
≡
t
{\displaystyle x\equiv t}
P
≡
T
{\displaystyle P\equiv T}
φ
:=
ω
t
{\displaystyle \varphi :=\omega t}
T
:=
2
π
{\displaystyle T:=2\pi }
x
:=
t
/
T
{\displaystyle x:=t/T}
T
:=
1
{\displaystyle T:=1}
Meistens ist
T
:=
2
π
{\displaystyle T:=2\pi }
t
0
:=
−
T
/
2
{\displaystyle t_{0}:=-T/2}
⟨
f
,
g
⟩
=
1
2
π
∫
−
π
π
f
(
t
)
¯
g
(
t
)
d
t
{\displaystyle \langle f,g\rangle ={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{\overline {f(t)}}\,g(t)\,\mathrm {d} t}
Das Skalarprodukt induziert die Fourier-Norm
‖
f
‖
:=
⟨
f
,
f
⟩
.
{\displaystyle \|f\|:={\sqrt {\langle f,f\rangle }}.}
Die Definition des Skalarproduktes ist so gewählt, dass es sich bei
‖
f
‖
{\displaystyle \|f\|}
Effektivwert von
f
{\displaystyle f}
Die Norm induziert die Fourier-Metrik
d
(
f
,
g
)
:=
‖
f
−
g
‖
{\displaystyle d(f,g):=\|f-g\|}
Die Funktionen
b
k
(
t
)
=
e
k
i
ω
t
{\displaystyle b_{k}(t)=\mathrm {e} ^{k\mathrm {i} \omega t}}
mit
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
B
=
{
b
k
∣
k
∈
Z
}
.
{\displaystyle B=\{b_{k}\mid k\in \mathbb {Z} \}.}
Mit
ω
:=
2
π
T
{\displaystyle \omega :={\tfrac {2\pi }{T}}}
allgemein
T
=
2
π
,
t
0
=
−
π
{\displaystyle T=2\pi ,\,t_{0}=-\pi }
a
k
[
f
]
=
2
T
∫
t
0
t
0
+
T
cos
(
k
ω
t
)
f
(
t
)
d
t
{\displaystyle a_{k}[f]={\frac {2}{T}}\int _{t_{0}}^{t_{0}+T}\cos(k\omega t)f(t)\,\mathrm {d} t}
a
k
[
f
]
=
1
π
∫
−
π
π
cos
(
k
t
)
f
(
t
)
d
t
{\displaystyle a_{k}[f]={\frac {1}{\pi }}\int _{-\pi }^{\pi }\cos(kt)f(t)\,\mathrm {d} t}
k
≥
0
{\displaystyle k\geq 0}
b
k
[
f
]
=
2
T
∫
t
0
t
0
+
T
sin
(
k
ω
t
)
f
(
t
)
d
t
{\displaystyle b_{k}[f]={\frac {2}{T}}\int _{t_{0}}^{t_{0}+T}\sin(k\omega t)f(t)\,\mathrm {d} t}
b
k
[
f
]
=
1
π
∫
−
π
π
sin
(
k
t
)
f
(
t
)
d
t
{\displaystyle b_{k}[f]={\frac {1}{\pi }}\int _{-\pi }^{\pi }\sin(kt)f(t)\,\mathrm {d} t}
k
≥
1
{\displaystyle k\geq 1}
Formeln für symmetrische Funktionen:
Bedingung
f
(
−
x
)
=
f
(
x
)
{\displaystyle f(-x)=f(x)}
b
k
=
0
{\displaystyle b_{k}=0}
a
k
=
4
T
∫
0
T
/
2
cos
(
k
ω
t
)
f
(
t
)
d
t
{\displaystyle a_{k}={\frac {4}{T}}\int _{0}^{T/2}\cos(k\omega t)f(t)\,\mathrm {d} t}
f
(
−
x
)
=
−
f
(
x
)
{\displaystyle f(-x)=-f(x)}
a
k
=
0
{\displaystyle a_{k}=0}
b
k
=
4
T
∫
0
T
/
2
sin
(
k
ω
t
)
f
(
t
)
d
t
{\displaystyle b_{k}={\frac {4}{T}}\int _{0}^{T/2}\sin(k\omega t)f(t)\,\mathrm {d} t}
Die Operationen
a
k
[
f
]
{\displaystyle a_{k}[f]}
b
k
[
f
]
{\displaystyle b_{k}[f]}
Additivität
Homogenität
(
λ
∈
R
)
{\displaystyle (\lambda \in \mathbb {R} )}
a
k
[
f
±
g
]
=
a
k
[
f
]
±
a
k
[
g
]
{\displaystyle a_{k}[f\pm g]=a_{k}[f]\pm a_{k}[g]}
a
k
[
λ
f
]
=
λ
a
k
[
f
]
{\displaystyle a_{k}[\lambda f]=\lambda a_{k}[f]}
b
k
[
f
±
g
]
=
b
k
[
f
]
±
b
k
[
g
]
{\displaystyle b_{k}[f\pm g]=b_{k}[f]\pm b_{k}[g]}
b
k
[
λ
f
]
=
λ
b
k
[
f
]
{\displaystyle b_{k}[\lambda f]=\lambda b_{k}[f]}
allgemein
T
=
2
π
,
t
0
=
−
π
{\displaystyle T=2\pi ,\,t_{0}=-\pi }
c
k
[
f
]
=
1
T
∫
t
0
t
0
+
T
e
−
k
i
ω
t
f
(
t
)
d
t
{\displaystyle c_{k}[f]={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}\mathrm {e} ^{-k\mathrm {i} \omega t}\,f(t)\,\mathrm {d} t}
c
k
[
f
]
=
1
2
π
∫
−
π
π
e
−
k
i
t
f
(
t
)
d
t
{\displaystyle c_{k}[f]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\mathrm {e} ^{-k\mathrm {i} t}\,f(t)\,\mathrm {d} t}
Kurz:
c
k
[
f
]
=
⟨
b
k
,
f
⟩
{\displaystyle c_{k}[f]=\langle b_{k},f\rangle }
Bei
c
k
[
f
]
{\displaystyle c_{k}[f]}
c
k
[
f
±
g
]
=
c
k
[
f
]
±
c
k
[
g
]
,
{\displaystyle c_{k}[f\pm g]=c_{k}[f]\pm c_{k}[g],}
c
k
[
λ
f
]
=
λ
c
k
[
f
]
(
λ
∈
C
)
.
{\displaystyle c_{k}[\lambda f]=\lambda c_{k}[f]\quad (\lambda \in \mathbb {C} ).}
Umrechnung zwischen komplexen und reellen Koeffizienten
Bearbeiten
reell zu komplex
komplex zu reell
c
0
=
1
2
a
0
{\displaystyle c_{0}={\frac {1}{2}}a_{0}}
a
0
=
2
c
0
{\displaystyle a_{0}=2c_{0}}
c
k
=
1
2
(
a
k
−
b
k
i
)
{\displaystyle c_{k}={\frac {1}{2}}(a_{k}-b_{k}\mathrm {i} )}
a
k
=
c
k
+
c
−
k
{\displaystyle a_{k}=c_{k}+c_{-k}}
c
−
k
=
1
2
(
a
k
+
b
k
i
)
{\displaystyle c_{-k}={\frac {1}{2}}(a_{k}+b_{k}\mathrm {i} )}
b
k
=
(
c
k
−
c
−
k
)
i
{\displaystyle b_{k}=(c_{k}-c_{-k})\mathrm {i} }
Alle Formeln gelten für
k
≥
1
{\displaystyle k\geq 1}
Für
m
,
n
≥
0
{\displaystyle m,n\geq 0}
∫
−
π
π
cos
(
m
t
)
sin
(
n
t
)
d
t
=
0.
{\displaystyle \int _{-\pi }^{\pi }\cos(mt)\sin(nt)\,\mathrm {d} t=0.}
Für
m
,
n
≥
1
{\displaystyle m,n\geq 1}
∫
−
π
π
cos
(
m
t
)
cos
(
n
t
)
d
t
=
π
δ
m
n
{\displaystyle \int _{-\pi }^{\pi }\cos(mt)\cos(nt)\,\mathrm {d} t=\pi \delta _{mn}}
und
∫
−
π
π
sin
(
m
t
)
sin
(
n
t
)
d
t
=
π
δ
m
n
.
{\displaystyle \int _{-\pi }^{\pi }\sin(mt)\sin(nt)\,\mathrm {d} t=\pi \delta _{mn}.}
Für
m
,
n
∈
Z
{\displaystyle m,n\in \mathbb {Z} }
1
T
∫
t
0
t
0
+
T
e
−
m
i
ω
t
e
n
i
ω
t
d
t
=
δ
m
n
{\displaystyle {\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}\mathrm {e} ^{-m\mathrm {i} \omega t}\mathrm {e} ^{n\mathrm {i} \omega t}\,\mathrm {d} t=\delta _{mn}}
wobei
δ
m
n
{\displaystyle \delta _{mn}}
Kronecker-Delta ist. Kurz:
⟨
b
m
,
b
n
⟩
=
δ
m
n
.
{\displaystyle \langle b_{m},b_{n}\rangle =\delta _{mn}.}
Fourier-Polynom:
p
n
(
t
)
=
a
0
2
+
∑
k
=
1
n
[
a
k
cos
(
k
ω
t
)
+
b
k
sin
(
k
ω
t
)
]
.
{\displaystyle p_{n}(t)={\frac {a_{0}}{2}}+\sum _{k=1}^{n}[a_{k}\cos(k\omega t)+b_{k}\sin(k\omega t)].}
Ist
f
{\displaystyle f}
t
{\displaystyle t}
f
(
t
)
=
lim
n
→
∞
p
n
(
t
)
.
{\displaystyle f(t)=\lim _{n\to \infty }p_{n}(t).}
Für
f
∈
L
2
(
−
π
,
π
)
{\displaystyle f\in L^{2}(-\pi ,\pi )}
lim
n
→
∞
‖
p
n
−
f
‖
=
0.
{\displaystyle \lim _{n\to \infty }\|p_{n}-f\|=0.}
Fourier-Polynom:
p
n
(
t
)
=
∑
k
=
−
n
n
c
k
e
k
i
ω
t
.
{\displaystyle p_{n}(t)=\sum _{k=-n}^{n}c_{k}\,\mathrm {e} ^{k\mathrm {i} \omega t}.}
Abstrakte Darstellung: Für
f
∈
L
2
(
−
π
,
π
)
{\displaystyle f\in L^{2}(-\pi ,\pi )}
f
=
∑
b
∈
B
⟨
b
,
f
⟩
b
.
{\displaystyle f=\sum _{b\in B}\langle b,f\rangle \,b.}
B
{\displaystyle B}
L
2
(
−
π
,
π
)
{\displaystyle L^{2}(-\pi ,\pi )}
![{\displaystyle a_{k}[f]={\frac {2}{T}}\int _{t_{0}}^{t_{0}+T}\cos(k\omega t)f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ec83d6c1cf58ff54f38f18c5c9825cc0078275)
![{\displaystyle a_{k}[f]={\frac {1}{\pi }}\int _{-\pi }^{\pi }\cos(kt)f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f99d59fa89d179d7d914dfab1f66daae0d4fa98)
![{\displaystyle b_{k}[f]={\frac {2}{T}}\int _{t_{0}}^{t_{0}+T}\sin(k\omega t)f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7801d29df38bb062bf29a5da700e0da82e069725)
![{\displaystyle b_{k}[f]={\frac {1}{\pi }}\int _{-\pi }^{\pi }\sin(kt)f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acb9bc7640f395e20b4fdb369ec7fd2fe22f3f13)
![{\displaystyle a_{k}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb04f56fff17eaea3b04beb782421695da8af2f)
![{\displaystyle b_{k}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c4d90727ddd964b93895f0c7ba90313f91b5cb)
![{\displaystyle a_{k}[f\pm g]=a_{k}[f]\pm a_{k}[g]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/485973b18202ee9f5a827b2619eb9ba8632fca76)
![{\displaystyle a_{k}[\lambda f]=\lambda a_{k}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f01caebfee472eddc283928b84be6e3936736f11)
![{\displaystyle b_{k}[f\pm g]=b_{k}[f]\pm b_{k}[g]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/931645b5592775cff35cef2920923c85988d4246)
![{\displaystyle b_{k}[\lambda f]=\lambda b_{k}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a302d1d2ce7c1787877ce31d0cb385dcaf67fb)
![{\displaystyle c_{k}[f]={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}\mathrm {e} ^{-k\mathrm {i} \omega t}\,f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b53975a510873af84f516f592dade6314444a64)
![{\displaystyle c_{k}[f]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\mathrm {e} ^{-k\mathrm {i} t}\,f(t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0008f2e979f45815c4d420adc2ed61b928df5fd0)
![{\displaystyle c_{k}[f]=\langle b_{k},f\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/415c9281208fff8b4bf2f7f80697549c8d95a86d)
![{\displaystyle c_{k}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28c60bb79d9226dc4be05c7e5419203f1a2eb9b0)
![{\displaystyle c_{k}[f\pm g]=c_{k}[f]\pm c_{k}[g],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c172bbc65dfaf341b748661aadec0a5c966c7bf)
![{\displaystyle c_{k}[\lambda f]=\lambda c_{k}[f]\quad (\lambda \in \mathbb {C} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d9b6cb57fdfb74c6cfb52ba72310caec85e047)
![{\displaystyle p_{n}(t)={\frac {a_{0}}{2}}+\sum _{k=1}^{n}[a_{k}\cos(k\omega t)+b_{k}\sin(k\omega t)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24729c53638a075d3e0c15d95ea90e52db3e485f)
