207:prolongements de fonctions. Exemples et

E F
A
¯
A
1/x R
sin(1/x)
A
F, d x ∈E/A
f A
f(x)−f(y)/x −y f C1
C1
a f0(x0) = l
C1
C∞(R)
u(k)(0) = ak
C∞[a, b]
b a C∞R
f g f(x) = g(x2)
g C∞0g(x)/x
C∞g C∞
R
C∞
C∞

x0=f(t, x)f
f
f
Ω\z0f(z) = o(1/(z−z0)
z0Ω
Ω\z0z0
Ω
Ω Ω
0
F(z) = RRezxe−x2dx
f(z) := π2cos(πz)/sin2(πz) = P∞
−∞ (−1)n/(z−n)2C\Z
ΓζC\ −N
C\1
f(1/2n) = f(1/(2n+ 1)) = 1/n
f(1/n) = f(−1/n)=1/n2x2
log(1/(1 −z) = Pzn/n 1
P12
k

R
a
a
a sup(f(k)(b/2)ρk
a/k!6C
λnλn+1/λn>a > 1
Panzλn
ε0= 1 ε2n=εnε2n+1 =−εnPεnzn
Pzn/n2
z0CV U
∆(z0, ρ, ϕ) = z|z=z0(1 −reiθ,06r6ρ, |θ|6ϕ06ϕ6π/2
06ρ < 2cos(ϕ)
P∞
1sin(nt)/n = (π−t)/2P(−1)n/(n+ 1) = ln(2)
Ω
U¯ Ω
¯
U
U αn→β
UΓαna
U[0,1[
f
Ωf
f(β)=1 β0f
fβ)6=f(β0)
¯
Ω¯ U
E F F X
E X f E

K Y
E F X E
X F |||f|||
f E |||f|||
L1(Rn)∩L2(Rn)L2(Rn)
L2
A L(H)
A A
Rg(x)6p(x)
g g
||x||
||f(x)|| =||x||2
||x|| =sup(|f(x)|) = max(|f(x)|)f
Jb(ϕ) = ϕ(b)|||Jb||| =||b|| Jb
b7→ JbE
E00
ϕ ϕ1,...ϕr∩Ker(ϕi)⊂
Ker(ϕ)∃λ1, . . . , λrϕ=Pλiϕi
(E, d)A E
f A A E ¯ f
E¯ f f A
X V
X K V f ∈Cc(X) 1
K1X
X V1,...Vn
X K V1∪. . .∪VnX
hihiVih1(x). . . +hn(x)=1
K

[0,1] 1 A0B
K(xi, yi)xi
n−1K[X]P(xi) = yi
P K[X]
∀x∈[a, b],∃ξ∈
I/f (x)−Pn(x) = fn+1(ξ)
(n+ 1)!
n
Y
i=0
(x−xi)
(Ω, µ)T
Lp(Ω) Lq(Ω) Lp0(Ω) Lq0(Ω) M0M1T
Mθ6Mθ
0M1−θ
1