PC* Espaces des fonctions continues admettant 2π comme période

π
C2πf
2π
∀α∈R,∀f∈ C2π, f([α, α + 2π]) = f([0,2π]) = f(R)
C2π
f7→ sup
[0,2π]
|f|=||f||[0,2π]
∞C2π
||f||∞
nZen:t7→ eint = (eit)n||en||∞= 1
g:t7→ cos(t) + sin(t)C2π||g||∞
C2π
∀(f, g)∈ C2
2π,⟨f, g⟩=1
2π2π
0
f(t)g(t)dt
||f||2=1
2π2π
0
|f(t)|2dt ||f||2≤ ||f||∞
f∈ C2π
∀t∈[−π, π], f(t) = |t| ∥f||2
∀(n, p)∈Z2,⟨en, ep⟩=1
2π2π
0
e−inteiptdt =δn,p
p∈NV ect(ep, e−p+1, ..., e1, e0, e1, ..., ep)
2p+ 1 Pp
P0⊂P1⊂... ⊂Pp⊂Pp+1 ⊂...
t7→ cos(t) + 2 sin(3t)P3
f N ∈N
f∈PNf N ∈N
(αk)−N≤k≤N∈C2N+1 ∀t∈R, f(t) =
N
k=−N
αkeikt
Pp
f∈ C2πPpf
PpSp(f) =
k=p
k=−p
⟨ek, f⟩ekPp

f
f
∀f∈ C2π,∀n∈Z,⟨en, f⟩=cn(f) = ˆ f(n) = 1
2π2π
0
e−intf(t)dt
f∈ C2π
f(t) = |t|t∈[−π, π]
2π2π
2π
F π , ∀α∈R,2π
0
F(t)dt =α+2π
α
F(t)dt
f
Sp(f)p
∀k∈N∗, gk=c−k(f)e−k+ck(f)ekg0=c0(f)e0, c0(f)
∀p∈N,∀t∈R, Sp(f)(t) = c0(f) +
p
k=1 c−k(f)e−ikt +ck(f)eikt
gkf
k∈Z
ck(f)ek
C2π
fR C 2π
p∈NSp(f)
∀p∈NSp(f) =
p
k=−p
ck(f)ekek:t7→ eikt
f f
lim
p→+∞||Sp(f)−f||2= 0
1
2π2π
0
|f(t)|2dt =||f||2
2=
+∞
−∞
|ck(f)|2

◦
εC2π
◦
t2[−π, π]
1/n4
CM
2π
2π
f∈ CM
2π
n∈Zcn(f) = ˆ f(n) = 1
2π2π
0
e−intf(t)dt
∀f∈ CM
2π,|c0(f)|2+
+∞
k=1
|c−k(f)|2+
+∞
k=1
|ck(f)|2=1
2π2π
0
|f(t)|2dt
∀f∈ CM
2π,∀p∈N,
k=p
k=−p
|ck(f)|2≤1
2π2π
0
|f(t)|2dt
◦ ∀f∈ CM
2π|ck(f)|2|c−k(f)|2
lim
k→+∞|ck(f)|= lim k→+∞|c−k(f)|= 0
◦ ∀(f, g)∈(CM
2π)2∀α∈C∀k∈Z
\
f+αg(k) = ckf+αg =ˆ f(k) + αˆg(k) = ck(f) + αck(g)
◦f∈ CM
2πa∈Rg∀t∈Rg(t) = f(t+a)
∀k∈Zck(g) = eikack(f)
◦f2π C1
∀k∈Zck(f′) = (ik)ck(f)
CNCN+1
f f
C∞
◦f g C2π
f=g⇔ ∀k∈Z, ck(f) = ck(g)
◦
∀(f, g)∈(C2π)2,⟨f, g⟩=
+∞
k=−∞
ck(f)ck(g) = 1
2π2π
0
f(t)g(t)dt

fR C 2π c1
fRf
t
f(t) =
+∞
−∞
cn(f)eint =a0(f)
2+
+∞
n=1
(an(f) cos(nt) + bn(f) sin(nt))
cn(f)
an(f)bn(f)
fR C 2π
C1
fR
x
f(x+) + f(x−)
2=
+∞
−∞
cn(f)einx =a0(f)
2+
+∞
n=1
(an(f) cos(nx) + bn(f) sin(nx))
◦f(x+)f(x−)
◦f
◦f∈ CM
2πf(0) = 0 ∀t∈]0,2π[f(t) = π−t
2