Corrigé I - al9ahira

f g E
V(f)−V∗(f)f[0, π/2] f
V(f)+V∗(f) : x7→ π/2 V(f)(0) = 0 = V∗(f)(π/2)
C1V(f)−V∗(g)
hV(f), gi=Zπ/2
0
V(f)(x)g(x)dx=V(f)(x)(−V∗(g)(x))π/2
0
+Zπ/2
0
f(x)V∗(g)(x)dx
hV(f), gi=hf, V∗(g)i
V∗◦V
f g E
hV∗◦V(f), gi=hV(f),V(g)i=hf, V∗◦V(g)i
hV∗◦V(f), fi=0=⇒f= 0E
hV∗◦V(f), fi= 0 0 = hV(f),V(f)i=kV(f)k2
V(f) = 0Ef= V(f)0= 0E
V∗◦V
λV∗◦Vfλλ
hV∗◦V(fλ), fλi=kV(fλ)k2
V(fλ)6= 0Efλ6= 0EV(fλ)0=fλ
hV∗◦V(fλ), fλi>0
hV∗◦V(fλ), fλi=hλfλ, fλi=λkfλk2
kfλk2>0 V∗◦V
fλ=1
λV∗◦V(fλ) = 1 λV∗(V(fλ))
f0
λ=1
λV(fλ)f00
λ=1
λfλfλC2
fλ(π/2) = 1
λV∗(V(fλ))(π/2) = 0 f0
λ(0) = 1
λV(fλ)(0)
fλy00 +1
λy= 0 y(π/2) = 0 y0(0) = 0
=⇒λV∗◦Vfλλ.
λ > 0 A B ∈R
fλ:x7→ A cos( x
√λ) + B sin( x √λ)f0
λ:x7→ − A
√λsin( x √λ) + B √λcos( x √λ)
0 = fλ(π/2) = A cos π
2√λ+ B sin π 2√λ0 = f0(0) = B
√λ
B=0 fλ:x7→ A cos x
√λ
fλ∈vect x7→ A cos x
√λ
A cos π
2√λ=fλ(π/2) = 0
fλ6= 0EA6= 0 cos π
2√λ= 0

k∈Zπ
2√λ=π
2+kπ =(2k+ 1)π 2
2√λ=(2k+ 1)
2
n∈Nλ=1
(2n+ 1)2n∈N2n+ 1 = |2k+ 1|
=⇒λ∈Rn∈Nλ=1
(2n+ 1)2f:x7→ cos( x
√λ) = cos((2n+ 1)x)
x∈[0,π
2]
V(f)(x) = Zx
0
cos((2n+ 1)t)dt=sin((2n+ 1)t)
2n+ 1 x
0=sin((2n+ 1)x)
2n+ 1
(V∗◦V) (f)(x) = Zπ/2
x
sin((2n+ 1)t)
2n+ 1 dt=−cos((2n+ 1)t) (2n+ 1)2π/2
x=cos((2n+ 1)x)
(2n+ 1)2−cos((2n+ 1)π/2) (2n+ 1)2
(V∗◦V) (f)(x) = λf(x)
(V∗◦V) (f) = λf f 6= 0E
λV∗◦Vn∈Nλ=1
(2n+ 1)2
E1/(2n+1)2(V∗◦V) = vect x7−→ cos((2n+ 1)x)
Snn x ∀k∈[[0, n]],P(Sn=k) = n
kxk(1 −x)n−k
0< k 6n kn
k=kn!
(n−k)!k!=n(n−1)! (n−1−(k−1))!(k−1)! =nn−1
k−1
E(Sn) =
n
X
k=0
kP(Sn=k) = 0 +
n
X
k=1
kn
kxk(1 −x)n−k=n
n
X
k=1 n−1
k−1xk(1 −x)n−k
E(Sn) = nx
n−1
X
j=0 n−1
jxj(1 −x)n−1−jj=k−1
n>2E(S2
n−Sn) =
n
X
k=0
(k2−k)P(Sn=k)
E(S2
n−Sn) =
n
X
k=2
k(k−1)n
kxk(1 −x)n−k=
n
X
k=2
n(n−1)n−2
k−2xk(1 −x)n−k
E(S2
n−Sn) = n(n−1)x2
n−2
X
j=0 n
jxj(1 −x)n−2−j=n(n−1)x2n= 1
E(S2
n) = E(S2 n−Sn) + E(Sn) = n(n−1)x2+nx
V(Sn) = E(S2
n)−E(Sn)2=n2x2−nx2+nx −(nx)2
E(Sn) = nx V(Sn) = nx(1 −x)
P(|Sn−E(Sn)|>nα)6
V(Sn)
(nα)2

V(Sn)
(nα)2=nx(1 −x) n2α2=x(1 −x) nα2
∀x∈[0,1],06x(1 −x)61/4
P(|Sn−E(Sn)|>nα)61
4nα2
(|Sn−E(Sn)|>nα) = (|Sn−nx|>nα) = Sn
n−x>α=[
06k6n
|k
n−x|>α
(Sn=k)
P(|Sn−E(Sn)|>nα) = X
06k6n
|k
n−x|>α
P(Sn=k)
X
06k6n
|k
n−x|>α
n
kxk(1 −x)n−k61 4nα2
Bn(f)(x) = E(f(Zn)) = E(f(Sn/n)) =
n
X
k=0
f(k
n)P(Sn=k)
Bn(f)(x) = n
kxk(1 −x)n−kf(k n)
n
X
k=0 n
kxk(1 −x)n−kf(x) = (x+ 1 −x)nf(x) = f(x)
Bn(f)(x)−f(x) =
n
X
k=0 n
kxk(1 −x)n−kf(k n)−f(x)
ε > 0
f[0,1]
kfk∞f[0,1]
f α > 0∀y, z ∈[0,1],|y−z|< α =⇒ |f(x)−f(y)|6ε
2
X
06k6n
|k
n−x|<α
n
kxk(1 −x)n−kf(k n)−f(x)
6X
06k6n
|k
n−x|<α
n
kxk(1 −x)n−k
f(k
n)−f(x)
6X
06k6n
|k
n−x|<α
n
kxk(1 −x)n−kε 2
6
n
X
k=0 n
kxk(1 −x)n−kε 2
X
06k6n
|k
n−x|<α
n
kxk(1 −x)n−kf(k n)−f(x)
6ε
2

X
06k6n
|k
n−x|>α
n
kxk(1 −x)n−kf(k n)−f(x)
6X
06k6n
|k
n−x|>α
n
kxk(1 −x)n−k
f(k
n)
+|f(x)|
62kfk∞X
06k6n
|k
n−x|>α
n
kxk(1 −x)n−k
X
06k6n
|k
n−x|>α
n
kxk(1 −x)n−kf(k n)−f(x)
6kfk∞
2nα2
lim
n→+∞kfk∞
2nα2= 0 N ∈N∀n∈N, n >N =⇒
kfk∞
2nα2
6ε
2
∀ε > 0,∃N∈N,∀n∈N, n >N =⇒(∀x∈[0,1],|Bn(f)(x)−f(x)|6ε)
kBn(f)−fk∞−→
n→+∞0
(Bn(f))n∈Nf[0,1]
V∗◦V(f)
p n p(X) =
n
X
n=0
akXkqk:t7→ cosk(t)
t7−→ p(cos(t)) vect(q0, . . . , qn)
k∈[[0, n]] t∈[0, π]
qk(t) = eit+ e−it
2k
=1
2k
n
X
j=0 n
je(2j−n)it=1
2kX
062j<n n
ke(2k−n)it+1
2kX
n<2j62nn
je(2j−n)it+r
r=((n
n/2)
2kn
0
p=n−kj 2j−n= 2n−2p−n=−(2p−n)
qk(t) = 1
2kX
06j<n/2n
ke(2k−n)it+1
2kX
06p<n/2n
n−pe−(2p−n)it+r=1
2k−1X
06k<n/2n
kcos((2k−n)t) + r
qk∈vect(c0, c1, . . . , ck)⊂vect(c0, c1, . . . , cp)
t7−→ p(cos(t)) [0, π] Fn

p k ∈Nt∈[0, π]cp(t)ck(t) = cos((p+k)t) + cos((p−k)t)
2
k6=p p +k > 0
hcp, ckiG=Zπ
0
cos((p+k)t) + cos((p−k)t)
2dt=sin((p+k)t) 2(p+k)+sin((p−k)t) 2(p−k)t=π
t=0
= 0
kcpk2=hcp, cpiG=Zπ
0
cos(2pt)+1
2dt
p= 0 kc0k2=Zπ
0
1 = π
p6= 0 kcpk2=sin(2pt)
4p+t 2t=π
t=0
=π
2
αn=(1
√πn= 0
q2
π
(αn)n∈N∈]0,+∞[N(αncn)n∈N
f∈Gf∈vect(αncn)n∈NGk·kG
Arccos t∈[0, π]7→ cos(t)∈[−1,1]
g=f◦Arccos [−1,1] Arccos
(gk)k∈N
g[−1,1]
N∞[−1,1] N∞(g−gk)−→
k→+∞0
k∈Nfk:[0, π]−→ R
t7−→ gk(cos(t))
N∈Nfk∈FNfk∈vect(αncn)n∈Nαn6= 0
t∈[0, π]|f(t)−fk(t)|=|f(Arccos(cos t)) −fk(Arccos(cos t))|=|g(cos(t)) −gk(cos(t))|
|f(t)−fk(t)|6N∞(g−gk)
kf−fkk2
G=Zπ
0
(f(t)−fk(t))2dt6Zπ
0
N∞(g−gk)2dt6πN∞(g−gk)2
kf−fkkG6√πN∞(g−gk)
kf−fkkG−→
k→+∞0
vect(αncn)n∈NfG
h·,·iG
vect(αncn)n∈NG
(αncn)n∈NG
(Fn)
n∈NFnFn+1
kf−PFn(f)kG= inf
g∈Fnkf−gkG>inf g∈Fn+1 kf−gkG=
f−PFn+1 (f) G>0
(kf−PFn(f)kG)n∈N
ε > 0
(fk) vect(αncn)n∈Nfk · kG
m∈Nkf−fmkG6ε
fm∈vect(αncn)n∈NN∈Nfm∈vect(αncn)06n6N