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TSI2
no8
ϕ:R→R
∀t∈R, ϕ(t) =  sin t
2
ϕ[−2π, 2π]
10 π
1
ϕRt7→ |t|t7→ sin t
2
ϕR
ϕC1]−π, 0[∪]0, π[
∀t∈]−π, 0[, ϕ(t) = −sin t
2∀t∈]0, π[, ϕ(t) = sin t 2
∀t∈]−π, 0[, ϕ0(t) = −1
2cos t 2∀t∈]0, π[, ϕ0(t) = 1 2cos t 2
ϕ00, π −π
ϕ[0,2π]C1[0,2π]
ϕ2π
ϕC1R
lim
t→0
t<0
ϕ0(t) = −1
26= lim
t→0
t>0
ϕ0(t) = 1
2
ϕC1R
ϕ
∀n∈N, bn= 0

ann= 0
a0=1
2πZπ
−π
ϕ(t)t=1
πZπ
0
ϕ(t)t
=1
πZπ
0
sin t
2t=2 π−cos t 2π
0
a0=2
π
n>1
an=1
πZπ
−π
ϕ(t) cos(nt)t=2
πZπ
0
ϕ(t) cos(nt)t
=2
πZπ
0
sin t
2cos(nt)t=1 πZπ
0sin (2n+ 1)t
2−sin (2n−1)t 2 t
=1
π−2 2n−1cos (2n−1)t 2+2 2n+ 1 cos (2n+ 1)t 2π
0
an=1
π2 2n−1−2 2n+ 1=2 π×2n−1−(2n+ 1) (2n−1)(2n+ 1)
∀n∈N, an=−4
π(4n2−1)
ϕC1
f f
∀t∈R, f(t) = 2
π−4 π
+∞
X
n=1
1
4n2−1cos(kt)
∞
X
n=1
1
4n2−1
∞
X
n=1
1
(4n2−1)2
f(0) = 0 = 2
π−4 π
+∞
X
n=1
1
4n2−1
+∞
X
n=1
1
4n2−1=1 2

TSI2
ϕ
2π
a2
0+1
2
+∞
X
k=1
a2
k=1
2πZπ
−π
ϕ2(t)t
1
2πZπ
−π
ϕ2(t)t=1
πZπ
0
sin2(t/2) t=1
2πZπ
0
(1 −cos(t)) t=1
2
4
π2+8 π2
+∞
X
n=1
1
(4n2−1)2=1 2
+∞
X
n=1
1
(4n2−1)2=π2 16 −1 2
ψR3[X]2R
∀P, Q ∈R3[X]2ψ(P, Q) =
3
X
i=0
P(i)Q(i).
ψ
P∈R3[X]ψ(P, P ) =
3
X
i=0
P(i)2= 0
∀i∈[[0,3]] , P (i)2= 0
P(0) = P(1) = P(2) = P(3) = 0 P
3
ψ
ψR3[X]
F=R2[X]B= (1, X, X2)

ψ(1,1) =
3
X
i=0
1 = 1 + 1 + 1 + 1 = 4 ψ(X, X2) =
3
X
i=0
i3= 36
ψ(1, X) =
3
X
i=0
i= 0 + 1 + 2 + 3 = 6 ψ(X2, X2) =
3
X
i=0
i4= 98
ψ(1, X2) = ψ(X, X) =
3
X
i=0
i2= 14
B0={P0, P1, P2}F
∀k∈ {0,1,2}Vect(P0, . . . , Pk) = Vect(1, . . . , Xk)ψ(Pk, Xk)>0.
P0, P1P2
(1, X, X2)
(Q0, Q1, Q2)
? Q0= 1
? Q1=X+λQ0
λ=−ψ(Q0, X)
ψ(Q0, Q0)=−ψ(1, X) ψ(1,1) =−6 4=−3 2
Q1=X−3
2
? Q2=X2+αQ1+βQ0
α=−ψ(Q1, X2)
ψ(Q1, Q1)=−ψ(X, X2)−3
2ψ(1, X2)
ψ(X, X)−3ψ(1, X) + 9
4ψ(1,1) =−15
5=−3
β=−ψ(Q0, X2)
ψ(Q0, Q0)=−ψ(1, X2) ψ(1,1) =−14 4=−7 2
Q2=X2−3X−3
2−7 2=X2−3X+ 1
? P0=Q0
pψ(Q0, Q0)=1
2
? P1=Q1
pψ(Q1, Q1)=√5
5X−3 2
? P2=Q2
pψ(Q2, Q2)
ψ(Q2, Q2) = ψ(Q2, X2+αQ1+βQ0) = ψ(Q2, X2)
=ψ(X2, X2)−3ψ(X, X2) + ψ(1, X2) = 98 −36 ×3 + 14 = 4

TSI2
P2=1
2(X2−3X+ 1)
P0=1
2;P1=√5 5X−3 2;P2=1 2(X2−3X+ 1)
(x0, x1, x2, x3) = (1,3,2,3)
Σ = (3
X
i=0 xi−P(i)2, P ∈F).
RR3[X]
∀i∈ {0,1,2,3}R(i) = xi
ϕ:R3[X]→R4
P7→ (P(0), P (1), P (2), P (3))
ϕ(P)=0⇒P(0) = P(1) = P(2) = P(3) = 0 ⇒P= 0
3
ϕ
ϕ
RR3[X]∀i∈ {0,1,2,3}R(i) = xi
F(P0, P1, P2)
F R F
p(R) = ψ(R, P0)P0+ψ(R, P1)P1+ψ(R, P2)P2
ψ(R, P0) = 1
2
3
X
i=0
R(i) = 1
2
3
X
i=0
xi=9
2
ψ(R, P1) = √5
5 3
X
i=0
iR(i)−3
2
3
X
i=0
R(i)!=√5
516 −27 2=√5 2
ψ(R, P2) = 1
2 3
X
i=0
i2R(i)−3
3
X
i=0
iR(i) +
3
X
i=0
R(i)!
=1
2(38 −48 + 9) = −1 2