Séries de Fourier

f:R→CT∀x∈R, f(x+T) = f(x)
f T
T
T[0, T [; [a, a +T[
-
6
a−Taa+T a + 2T
xPUn(x)
Un(x) = ancos(nx) + bnsin(nx)x∈R; (an, bn)∈C2
PUn(x)
Sn(x) = a0+ancos(x) + bnsin(x) + · · · +ancos(nx) + bnsin(nx)
Sn(x) 2π
n= 1 U1(x) = a1cos x+b1sin x2π
n > 1Un(x) = ancos(nx) + bnsin(nx)2π
n
PUn(x)R
Sn=P+∞
n=0 Un(x) = limn→+∞Sn(x) 2π
∀n≥0Un(x) = ancos(nx)+bnsin(nx)R
f(x)PUn(x)

PUn(x)Un(x) = ancos(nx) + bnsin(nx)
PanPbn
PUn(x)Rf(x)R
∀x∈R|Un(x)|=|ancos(nx) + bnsin(nx)| ≤ |an|+|bn|
| {z }
Vn
P|an|P|bn|PVn
PUn(x)R
PUn(x)R
∀n∈NUn(x)R
PUn(x)R
f(x)R
(an) (bn)Pancos(nx) + bnsin(nx)
R−2πZ
Ik= [2kπ +α, 2(k+ 1)π−α]
α∈]0, π[
∀x∈R−2πZPaneinx
Pancos(nx)Pansin(nx)Pbneinx
εn=anαn=einx εn&0
n
X
k=0
αk =
n
X
k=0
(eix)k =
1−(eix)n+1
1−eix ≤
1
1−eix +
ei(n+1)x
1−eix 
eix 6= 1 ⇒x= 2kπ x ∈R−2πZ
∀n, p, ∀x∈Ik|Sn+p(x)−Sn(x)| ≤ 2can+1 c=2
|1−eix|
PUn(x)
∀n∈NUn(x) = ancos(nx) + bnsin(nx)
Un(x) = aneinx +e−inx
2+bneinx −e−inx 2i=an−ibn 2einx +an+ibn 2e−inx

∀n≥1cn=an−ibn
2(1)
c−n=an+ibn
2(2)
∀n≥1Un(x) = cneinx +c−ne−inx PUn(x)
f(x)c0=a0
f(x) = c0+ (c1eix +c−1e−ix) + (c2ei2x+c−2e−i2x) + · · · + (cneinx +c−ne−inx) + · · · =
+∞
X
−∞
cneinx
anbncn
∀n≥1an=cn+c−n(3)
bn=i(cn−c−n) (4)
n= 0 a0= 2c0
c0=a0
2f(x)PUn(x)
f(x) = P+∞
−∞ cneinx
a0
2+a1cos x+b1sin x+· · · +ancos nx +bnsin nx
PUn(x)Rf(x) = P+∞
−∞ cneinx
Rcn
f(x)
p∈Z
f(x)e−ipx =e−ipx
+∞
X
n=−∞
cneinx ⇒f(x)e−ipx =
+∞
X
n=−∞
cnei(n−p)x
Pcnei(n−p)x
Z2π
0
f(x)e−ipxdx =Z2π
0
+∞
X
n=−∞
cnei(n−p)xdx =
+∞
X
n=−∞
cnZ2π
0
ei(n−p)xdx
n6=p, :Z2π
0
ei(n−p)xdx =1
i(n−p)ei(n−p)x2π
0
= 0
n=p, :Z2π
0
ei(n−p)xdx =Z2π
0
1dx = 2π
∀p∈ZZ2π
0
f(x)e−ipxdx =cp2π
cp=1
2πZ2π
0
f(x)e−ipxdx
PUn(x)
a0
2+a1cos x+b1sin x+· · · +ancos nx +bnsin nx
anbn
∀n∈Nan=cn+c−n
bn=i(cn−c−n)