Tabla.

Espacio de Hilbert
separable (H)
Boreliano (B)
Func. peso ω(x)
Conjunto lin. indep.
a ortonormalizar en L2 (B)
∞
Base ortonormal {eω
n }n=0
2
numerable en Lω (B)
H = L2 [−1, 1]
B = [−1, 1]
{xn }∞
n=0 ⊂ H
eω
n =
ω(x) = 1
p
n + 1/2Pn (x)
H = L2 [0, ∞]
{xn e−x/2 }∞
n=0 ⊂ H
B = [0, ∞]
ω(x) = e−x
(−1)n dn
n!2n dxn (1
{xn e−x
B=R
ω(x) = e−x
2
/2 ∞
}n=0
2
Ln (x) ≡ polinomio de Laguerre
L0 = 1,
1 x dn
−x n
x )
n! e dxn (e
(n!2n
1
√
π)1/2
B = [a, b]
i b−a x +∞
√1
}n=−∞
{en }+∞
n=−∞ = { b−a e
b − a = 2π
ω(x) = 1
inx +∞
√1
{en }+∞
}n=−∞
n=−∞ = { 2π e
⇒
R
B
dn −x2
dxn e
n hen , f i en
|f (x)|2 dx = ||f ||2 =
P
f = g ω 1/2 ∈ L2 (B) ⇔ g = f ω −1/2 ∈ L2ω (B)
c M.C. Boscá y E. Romera, Univ. Granada.
n
⇔ limn→∞ ||f −
|hen , f i|2
con
R
B
= 0 ⇔ ∀ǫ > 0
2
Pn (x)
dx
R +∞
−∞
2
e−x Hn (x)Hm (x)dx =
H0 (x) = 1,
R
B
n (x)
− 2x dPdx
+ n(n + 1)Pn (x) = 0
en (x) = e−x/2 Ln (x)
√
π2n n!δn,m
en (x) =
Hn+1 (x) = 2xHn (x) − 2nHn−1 (x),
d2 Hn (x)
dx2
Hn (x) = (−1)n Hn (−x),
∃m ∈ N :
n≥1
n≥1
n (x)
+ (1 − x) dLdx
+ nLn (x) = 0
(2)
n=0 hen , f i en ||
hen , f i =
Ln (x)
dx2
n + 1/2Pn (x)
(n + 1)Ln+1 (x) = (2n + 1 − x)Ln (x) − nLn−1 (x),
√1
{en }∞
n=0 = { 2π ,
Pm
(1 − x2 ) d
e−x Ln (x)Lm (x)dx = δn,m
√1
{en }∞
n=0 = { b−a ,
(1)
P
2
p
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x),
(2)
2πn
H = L2 ([a, b])
H = L2 (B)
xd
Hn (x)
2
2
2n+1 δn,m
Pn (−x) = (−1)n Pn (x),
0
Hn (x) = (−1)n ex
Convergencia en norma ∀f ∈ H : f =
P0 (x) = 1,
− x 2 )n
Hn (x) ≡ polinomio de Hermite
(1)
en (x) =
Pn (x)Pm (x)dx =
R∞
eω
n =
⊂H
R1
eω
n = Ln (x)
Ln (x) =
H = L2 (R)
∞
2
Base
R ∗ ortonormal {en }n=0 en L (B)
e (x)em (x)dx = δnm
B n
−1
Pn (x) ≡ polinomio de Legendre
Pn (x) =
2
Relación
R ω∗ deω ortonormalización en Lω (B)
e (x)em (x)ω(x)dx = δnm
B n
|f (x) −
q
q
Pm(ǫ)
n=0
1
π
2
b−a
n≥1
n (x)
− 2x dHdx
+ 2nHn (x) = 0
cos 2πkx
b−a ,
cos kx,
2
/2
e−x
√
H (x)
(n!2n π)1/2 n
q
1
π
q
2
b−a
+∞
sin 2πkx
b−a }k=1
sin kx}+∞
k=1
hen , f i en |2 dx < ǫ2 con {en }∞
n=0 ≡ b.o.n. de H
e∗n (x)f (x)dx
TABLA 1 (general)