Les fonctions presque périodiques

y00 +ω2y=g
g
• B R C
k k ∀f∈ B kfk= supx∈R|f(x)|(B,k k)
•f∈ B t∈Rf(.+t)∀x∈Rf(.+t)(x) = f(x+t)
Γf={f(.+t), t ∈R}f
f∈ B Γf
(B,k k) (an)n∈Nφ
(f(.+aφ(n)))n∈NR
E
f T Γf=
{f(.+t), t ∈[0, T ]}t7→ f(.+t)fΓf
Γff
f∈ E Γf

x7→ cos(x)+cos(√2x)
g
E(B,k k)
E B
(fn)n∈Nf ε > 0
mkfm−fk ≤ ε
3t1, . . . , tp(B,k k)fm(.+ti)
ε
3i∈ {1, . . . , p}ΓfmΓfmt∈Ri∈ {1, . . . , p}
kf(.+t)−f(.+ti)k ≤ kf(.+t)−fm(.+t)k+kfm(.+t)−fm(.+ti)k+kfm(.+ti)−f(.+ti)k
(B,k k)fm(.+ti)ε i ∈ {1, . . . , p}ΓfΓf
f∈ E E 
E B
x7→ cos(x) + cos(√2x)
g
g:x7→ P+∞
n=1 1
n2cos( x n)
R
f∈ E f t 7→ f(.+t)
0 (xn)n∈N0g(f(.+xn))n∈N
f f =g(f(.+xn))n∈N
Γff f
t7→ f(.+t) 0 
f T f
1
TRT
0f f R
f T +∞1
TRT
0f
f∈ E Γf
f

f∈ E c c0
Γfε > 0α1. . . αpβ1. . . βqu1. . . upv1. . . vq
kPp
i=1 αif(.+ui)−ck ≤ εkPq i=1 βif(.+vi)−c0k ≤ ε
g=X
1≤i≤p,1≤j≤q
αiβjf(.+ui+uj)
ε c c
c0|c−c0|≤ 2ε
CfΓfCf
T gT:x7→ 1
TRx+T
xf
CfgT1
nPn
k=1 f(.+kT
n)n≥1
Cf(un)n∈Ng(gun)n∈N
φ(guφ(n))n∈Ng g x y
guφ(n)(x)−guφ(n)(y)=
1
uφ(n)Zx+uφ(n)
x
f−Zy+uφ(n)
y
f =
1
uφ(n) Zy
x
f−Zy+uφ(n)
x+uφ(n)
f!≤2|y−x|kfk
uφ(n)
n+∞g(x) = g(y) (gun)n∈N
c Cfc
gTc T
+∞x1
TRx+T
xf T +∞
x
M(f) = lim
T→+∞
1
TZT
0
f
f
M∈ F(E,C)
f T0M(f) = 1
T0RT0
0f M
f
E
f∈ E M(f)≥0f
M(f)f∈ E x0
f(x0)>0ε=f(x0)
2>0t1,...tp(B,k k)f(.+ti)
ε i ∈ {1, . . . , p}Γfg=Pp
i=1 f(.+ti)M(g) = pM(f)x
i∈ {1, . . . , p} kf(.+x0−x)−f(.+ti)k ≤ ε f(x+ti)≥f(x0)
2g(x)≥f(x0) 2M(g)≥f(x0) 2
M(f)≥f(x0)
2p>0
E
E
f g hf, gi=M(fg)
h,ik k2
eλ
R C ∀x∈Reλ(x) = eiλx λ∈Reλ∈ E
(eλ)λ∈Rh,i

(eλ)λ∈R
f∈ E fˆ f∀λ∈R
ˆ
f(λ) = heλ, fi
f T λ /∈2π
TZˆ
f(λ)=0 n∈Zˆ f(2πn
T)
n f
(|ˆ f(λ)|2)λ∈Rkfk2
2
ˆ
fSp(f) = {λ∈R,ˆ f(λ)6= 0}ˆ f
f∈ E ˆ f f
f
KfEg∈ E x Kf(g)(x) = M(f(x−.)g)λ
Kf(eλ) = ˆ f(λ)eλ
(E,k k2) (E,k k2)f Kf
g:x7→ f(−x)Kf(g)(0) = M(|f|2)
Kf
λ eλ
(eλ)λ∈R(E,k k) (E,k k2)
(eλ)λ∈R
(E,k k2)f=Pλ∈Sp(f)ˆ f(λ)eλf
f=Pλ∈Sp(f)ˆ f(λ)eλ
(E,k k)
g g :x7→ P+∞
n=1 1
n2cos( x n)
g=1
2Pn∈Z∗
1
n2e1
n
∀λ∈Rˆg(λ) = (1
2n2λ=1 nn∈Z∗
0
Sp(g) = {1
n, n ∈Z∗}
f∈ E C1f0
λˆ f0(λ) = iλ ˆ f(λ)
f0
(n(f(.+1
n)−f))n≥1f0
Eˆ f0ˆ f
f∈ E F f F F
λ6= 0 ˆ F(λ) = ˆ
f(λ)
iλ
M(f)=0 F∈ E f=P+∞
n=1 1
n2e1
n2f∈ E M(f) = 0
F f En∈N∗ˆ F(1
n) = −i(|ˆ
F(λ)|2)λ∈R

f F F :x7→ Rx
0f
f x u F (x+u) = F(u) + Rx
0f(.+u)
f(f(.+un))n∈Ng
x7→ Rx
0f(.+un)RG
f∈ E F(un)n∈N(f(.+un))n∈N
g φ (F(uφ(n)))n∈N)lIm(F)
(F(.+uφ(n)))n∈NH=l+G
R
Im(F(.+u)) = Im(F)uIm(H)⊂Im(F) (g(.−un))n∈N
f f g K1=m+FIm(K1)⊂Im(G)
K2=K1+lIm(K2)⊂Im(H) Im(H)⊂Im(F)
Im(K2)⊂Im(H)⊂Im(F)F K2Im(F)F=K2H
g H (un)n∈N
H g Im(H) = Im(F)l
(F(un))n∈N
(f(.+un))n∈N(f(.+vn))n∈N
(F(un))n∈N(F(vn))n∈N
f∈ E F(un)n∈N
(f(.+un))n∈NRg
(F(un))n∈Nl(F(.+un))n∈NR
ΓF
(F(.+un))n∈Nl+G
R(F(.+un))n∈N
(pk)k∈N(qk)k∈N
(xk)k∈Nε0>0k|F(xk+upk)−F(xk+uqk)| ≥ ε0
(k)k∈N(g(.+xk))k∈Nh
(f(.+upk))k∈N(f(.+uqk))k∈Ng(f(.+xk+upk))k∈N
(f(.+xk+uqk))k∈Nh(F(xk+upk))k∈N
(F(xk+uqk))k∈Nk|F(xk+upk)−F(xk+uqk)| ≥
ε0ΓFF
n≥1
f(n)+
n−1
X
k=0
akf(k)=g
g∈ E akfR
g= 0 λ1. . . λpP=Xn+Pn−1
k=0 akXk
m1. . . mpf:x7→ Pp
k=1 pk(x)eλkxpk
mk−1Rλkpk6= 0
pk
g
R
n
R
R Cn
G:t7→ (g1(t), g2(t),...gn(t)) R CnG
i∈ {1,...n}gi∈ E
E