n - Les-Mathematiques.net

E E∗
E. E
n, n.
E n ≥1B=
(ei)1≤i≤nE. T={ϕ1,···, ϕn}
E E∗
G=
ϕ1(e1)ϕ1(e2)··· ϕ1(en)
ϕ2(e1)ϕ2(e2)··· ϕ2(en)
ϕn(e1)ϕn(e2)··· ϕn(en)
Mn(R).
B∗= (e∗
i)1≤i≤nB,
j1n
ϕj=
n
X
i=1
ϕj(ei)e∗
i,
tGB T
E∗tG
G
G= ((ϕi(ej)))1≤i,j≤n
n≥1n
ϕ1,··· , ϕnE,
y= (y1,··· , yn)Rnx E
∀i∈ {1,··· , n}, ϕi(x) = yi.

(6.1)
E, y RnT=
{ϕ1,··· , ϕn}E∗. y Rn,
(6.1) E.
B= (ei)1≤i≤nE,
x∈E x =
n
P
j=1
xjeji1n,
ϕi(x) =
n
X
j=1
ϕi(ej)xj.
(6.1) E y Rn
yRn
n
X
j=1
ϕi(ej)xj=yi(1 ≤i≤n)
(x1,··· , xn)Rn,
G= ((ϕi(ej)))1≤i,j≤n
TE∗.
G(6.2)
(6.1) E.
T= (ϕi)1≤i≤nE∗
i1n, LiE
∀j∈ {1,···, n}, ϕj(Li) = ½1j=i,
0j6=i.
L=
(Li)1≤i≤nE y Rn,
½x∈E,
ϕi(x) = yi(1 ≤i≤n),
x=
n
X
j=1
yjLj.
n
P
i=1
αiLi= 0, j
1n
αj=ϕjÃn
X
i=1
αiLi!= 0.
LE,

yRn, x =
n
P
j=0
yjLj
ϕi(x) = ϕiÃn
X
j=1
yjLj!=yi(1 ≤i≤n)
L
Rnu E Rn
∀x∈E, u (x) = 
ϕ1(x)
ϕn(x)
.
E=Rn[x]
n,
EB=
(ej)0≤j≤n, ej(x) = xj.
n+ 1
x0< x1<··· < xni0n,
ϕixiE=Rn[x]
∀P∈E, ϕi(P) = P(xi).
G= ((ϕi(ej)))0≤i,j≤n=
1x0··· xn
0
1x1··· xn
1
1xn··· xn
n
.
det (G) = Y
0≤i<j≤n
(xj−xi)
xi
(y0,··· , yn)Rn+1,
P
½P∈Rn[x],
P(xi) = yi(0 ≤i≤n).
(x0,··· , xn)
(y0,··· , yn). Li
Li(x) =
n
Y
j=0
j6=i
x−xj
xi−xj

P(x) =
n
X
i=0
yi
n
Y
j=0
j6=i
x−xj
xi−xj
.
a
i0n, ϕi
E=Rn[x]
∀P∈E, ϕi(P) = P(i)(a).
G= ((ϕi(ej)))0≤i,j≤n=³³e(i)
j(a)´´0≤i,j≤n=
1a··· an
0 1! ··· nan−1
0 0 ··· n!
.
e(i)
i(a) = i!,
det (G) =
n
Y
i=1
i!6= 0.
(y0,··· , yn)Rn+1,
P½P∈Rn[x],
P(i)(a) = yi(0 ≤i≤n).
a
(y0,··· , yn). Li
Li(x) = 1
i!(x−a)i
P(x) =
n
X
i=0
yi
i!(x−a)i.
n a
n.
n+ 1
x0< x1<··· < xn, α0, α1,··· , αn,
m=n+
n
X
i=0
αi
(i, k)i0n k
0αi, ϕi,k E=
Rm[x]
∀P∈E, ϕi,k (P) = P(k)(xi).

y= 0
P
½P∈Rm[x],
P(k)(xi) = 0 (0 ≤i≤n, 0≤k≤αi),
P
P(k)(xi)=0 k0αi
P(x−xi)αi+1 ,
P∈Rm[x],
P(x) = Q(x)
n
Q
i=0
(x−xi)αi+1
P
y= (y0,0,··· , y0,α0,··· , yn,0,··· , yn,αn)∈Rm+1
P
½P∈Rm[x],
P(k)(xi) = yi,k (0 ≤i≤n, 0≤k≤αi).
(x0,··· , xn),
(α0,··· , αn)y.
Hi,k i0
n k 0αi,
Hi,k ∈Rm[x],
H(q)
i,k (xp) = ½0 (p, q)6= (i, k),
1 (p, q) = (i, k),(0 ≤p≤n, 0≤q≤αp)
P=
n
X
i=0
αi
X
k=0
yi,kHi,k.
α= (0,··· ,0) ,
α= (1,···,1) ,
½Hi,0(x) = (1 −2L0
i(xi) (x−xi)) L2 i(x),
Hi,1(x) = (x−xi)L2
i(x)
Li
½P∈R2n+1 [x],
P(xi) = yi, P 0(xi) = 0 (0 ≤i≤n),
xiTn+1,