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Dn
C1
C1
C1
T > 0CT(R)TR
R
(f|g) = 1
TZT
0
f(t)g(t)t=1
TZT/2
−T/2
f(t)g(t)t
ω=2π
T
(f|g) = 1
TZT
0
f(t)g(t)t=1
TZT /2
0
f(t)g(t)t+1
TZT
T /2
f(t)g(t)t
u=t−T
[0, T ]
[−T/2, T /2]
F
ϕn:t7→ cos nωt n ∈Nψn:t7→ sin nωt n ∈N∗
(ϕ0|ϕ0)=1 p∈N∗(ϕp|ϕp)=(ψp|ψp) = 1
2
DnCT(R)ϕ0, ϕ1, . . . , ϕn, ψ1, . . . , ψn2n+ 1

Dn
DnCT(R)
CT(R) = Dn⊕D⊥
n
fCT(R)
f=
n
X
k=0
akϕk+
n
X
k=1
bkψk+h
a0, . . . , an, b1, . . . , bn∈Rh∈D⊥
n
akf ϕkbkf ψk
f∈ CT(R)
a0(f) = 1
TZT/2
−T/2
f(t)t
k∈N∗
ak(f) = 2
TZT/2
−T/2
f(t) cos kωt t bk(f) = 2
TZT/2
−T/2
f(t) sin kωt t
a0(f) = (f|ϕ0)k∈N∗ak(f) = 2(f|ϕk), bk(f) = 2(f|ψk)
Sn(f) =
n
X
k=0
ak(f)ϕk+
n
X
k=1
bk(f)ψk=a0(f)ϕ0+
n
X
k=1 ak(f)ϕk+bk(f)ψk
Dnf
akf bkf
[−T/2, T /2] [0, T ]
fx+T
2=−f(x)
f∈ CT(R)t
Sn(f)(t) = a0(f) +
n
X
k=1 ak(f) cos kωt +bk(f) sin kωt
n f t
Sn(f)Dnf
Sn(f)(t)f(t)n

T= 2π ω = 1
a0(f) = 1
2πZπ
−π
f(t)t
k∈N∗
ak(f) = 1
πZπ
−π
f(t) cos kt t bk(f) = 1
πZπ
−π
f(t) sin kt t
f:R7→ R2π
∀x∈[0, π], f(x)=1−2x
π
n DnSn(f)f Dn
flim
n→+∞kSn(f)k2=kfk2
f∈ CT(R)
a0(f)2+1
2
∞
X
k=1 ak(f)2+bk(f)2=1
TZT/2
−T/2
f(t)2t
f:R7→ R2π
∀x∈[0, π], f(x)=1−2x
π
∞
X
k=0
1
(2k+ 1)4
∞
X
n=1
1
n4
C1
[a, b] (a0, a1, . . . , an)
a=a0< a1<· · · < an=b
f: [a, b]→R(a0, a1, . . . , an) [a, b]
f]ai, ai+1[ [ai, ai+1]
f]ai, ai+1[ ]ai, ai+1[
f
(a0, a1, . . . , an)

f:I→R[a, b]I
I
I
f:x7→ 1x>0
−1x < 0
f: [0, π]7→ Rf(x) = tan x x 6=π/2
0x=π/2
f:R→RT[a, a +T]
Cm([a, b],R)Cm(I, R)Cm,T (R,R)
Cm([a, b],R)Cm(I, R)Cm,T (R,R)R
T[a, a +T]
T
T= 2 f(x)=1−x x ∈]−1,1]
T= 4 f f(x) = x3x∈[0,2]
T= 2π f f(π)=0 f(x) = cos x x ∈[0, π[
T= 2 f f(x) = exx∈[0,1]
T= 2 f f(−1) = f(0) = 0 f(x) = 1 x∈]−1,0[
f:x7→ x−E(x)
∀k∈N, E(x+k) = E(x) + k
f
fR
I
f, g ∈ Cm(I, R)α, β ∈Rf g I αf +βg
I
ZI
(αf +βg) = αZI f+βZI g

I J I ∩J
I∪J f I ∪J
ZI∪J
f=ZI f+ZJ f
f∈ Cm(I, R+)ZI f>0
f∈ C(I, R+)ZI f= 0
∀t∈I, f(t)=0.
f∈ Cm(I, R)
ZI
f
6ZI
|f|
[−1,1] f f(0) = 1
f(x) = 0 x∈[−1,0[∪]0,1]
f: [a, b]→R[a, b]C1]a, b]a f0λ f
C1[a, b]f0(a) = λ
fC1]a, b]f0(a) = λ
lim
x→a+
f(x)−f(a)
x−a=λ a lim
x→a+f0(x) = λ
lim
x→a+
f(x)−f(a)
x−a=λ
ε > 0α > 0
∀t∈]a, a +α],|f0(t)−λ|6ε
x∈]a, a +α]t∈]a, x[
f(x)−f(a)
x−a=f0(t)
f(x)−f(a)
x−a−λ
6ε
∀ε > 0,∃α > 0,∀x, x∈]a, a +α] =⇒
f(x)−f(a)
x−a−λ
6ε
lim
x→a+f0(x)=+∞
−∞ lim
x→a+
f(x)−f(a)
x−a= +∞ −∞ f
a
C1
f: ]a, b]→RC1fC1[a, b]ˆ f
[a, b]Rf]a, b]C1[a, b]
ˆ
f f a