Cours - Page Personnelle de Jérôme Von Buhren

T > 0ω=2π
T
f: [a, b]→R
C1[a, b]
a=a0<··· < an=b
f]ai, ai+1[
C1[ai, ai+1]
C1[a1, a2]
x=a1
C1
x
y
a a1a2b
••
•
•x
y
a a1a2b
••
••
f: [a, b]→R
T f :R→R
C1
C1
f:R→RT
f: [a, b]→R
f[a, b]
Zb
a
f(t)dt =
n−1
X
i=0 Zai+1
ai
f(t)dt.
1
x
y
a a1a2b

f: [a, c]→Ra<b<c
Zc
a
f(t)dt =Zb
a
f(t)dt +Zc
b
f(t)dt.
f, g : [a, b]→R(λ, µ)∈R2
Zb
a
(λf +µg)(t)dt =λZb
a
f(t)dt +µZb
a
g(t)dt.
f[a, b]Zb
a
f(t)dt >0
f6g[a, b]Zb
a
f(t)dt 6Zb
a
g(t)dt
f: [a, b]→R
f
x
y
ab
• •
T
f:R→Rn∈N∗
a0(f) = 1
TZT
0
f(t)dt,
an(f) = 2
TZT
0
f(t) cos(nωt)dt, bn(f) = 2
TZT
0
f(t) sin(nωt)dt.
anbn
T
T
a0f
2π f :R→R
f(t) = t t ∈[−π, π[
a0= 0, an= 0, bn= (−1)n+1 2
nn∈N∗.
f:R→RT
f n ∈N∗bn= 0
f n ∈Nan= 0
f x ∈Rf(x+T
2) = −f(x)
n∈N∗a0=a2n=b2n= 0

x
y
T
T/2
• • • x
y
T
T/2
n∈NT
f:R→R
Sn(f)(t) = a0+
n
X
k=1
(akcos(kωt) + bksin(kωt)) .
S3(f)(t) = 2 sin(t)−sin(2t) + 2
3sin(3t).
T
f:R→R
S(f)(t) = a0+
+∞
X
n=1
(ancos(nωt) + bnsin(nωt)) .
S(f)(t) =
+∞
X
n=1
(−1)n+1 2
nsin(nt).
CT(R,R)R
RT
(·|·) : CT(R,R)×CT(R,R)→R
∀(f, g)∈CT(R,R)2,(f|g)7→ 1
TZT
0
f(t)g(t)dt
CT(R,R)
(CT(R,R),(· | ·))
f∈CT(R,R)
kfk=s1
TZT
0
f(t)2dt.
(·|·)

t7→ 1, t 7→ √2 cos (ωkt), t 7→ √2 sin (ωkt)k∈N∗CT(R,R)
f∈CT(R,R)Sn(f)f
Pn,T = Vect{t7→ 1, t 7→ cos(kωt), t 7→ sin(kωt)|k∈J1, nK}.
f∈CT(R,R)
kSn(f)k2=a2
0+1
2
n
X
k=1 a2
k+b2 k.
f:R→RTR
Pa2
nPb2 n
1
TZT
0|f(t)|2dt =a2
0+1
2
+∞
X
n=1 a2
n+b2 n.
π2
3= 2
+∞
X
n=1
1
n2,
+∞
X
n=1
1
n2=π2 6.
f:R→R
˜
f:R→R
˜
f(t) = lim
h→0f(t+h) + f(t−h)
2.
f f =˜ f
f˜ f
x
y
•
•
f
x
y
•
•
˜
f

f:R→RTC1
Rf˜ f
∀t∈R, S(f)(t) = lim
n→+∞Sn(f)(t) = ˜
f(t).
C1
f:R→RT
˜
f(t)=2
+∞
X
n=1
(−1)n+1
nsin(nt).
f
x
y
π
−π
π
−π
• •
f
x
y
π
−π
π
−π
• •
˜
f
x
y
π
−π
π
−π
S3(f)
x
y
π
−π
π
−π
S8(f)