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)