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Approximación y Proyección en el Espacio de Hilbert

Module by: Justin Romberg. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Approximation and Projections in Hilbert Space by Justin Romberg

Summary: Este modulo introduce la aproximación y la proyección en el espacio de Hilbert.

Introducción

Dada una linea 'l' y un punto 'p' en el plano, ¿ Cuál es el punto más cercano 'm' a 'p' en 'l'?

Figura 1: Figura del punto 'p' y la linea 'l' mencionadas.
Figura 1 (approx_f1.png)

Mismo problema: Sea xx y vv vectores en R2 2 . Digamos ∥v∥=1 v 1 . ¿Para qué valor de αα es ∥x−α⁢v∥ 2 x α v 2 minimizado? (¿qué punto en el espacio generado{v} mejor se aproxima a xx?)

Figura 2:
Figura 2 (approx_f2.png)

La condición es que x− α ^ ⁢v x α ^ v y α⁢v α v sean ortogonales.

Calculando α

¿Cómo calcular α ^ α ^ ?

Sabemos que ( x− α ^ ⁢v x α ^ v ) es perpendicular para todo vector en el espacio generado {v}, así que ∀β,∀⁢β:〈x− α ^ ⁢v,β⁢v〉=0 β ∀ β x α ^ v β v 0 β¯⁢〈(x,v)〉− α ^ ⁢β¯⁢〈(v,v)〉=0 β x v α ^ β v v 0 por que 〈v,v〉=1 v v 1 , por lo tanto (〈(x,v)〉− α ^ =0)⇒( α ^ =〈x,v〉) x v α ^ 0 α ^ x v El vector más cercano en el espacio generado{v} = 〈(x,v)〉⁢v x v v , donde 〈(x,v)〉⁢v x v v es la proyección de xx sobre vv.

¿Punto a un plano?

Figura 3:
Figura 3 (approx_f3.png)

Podemos hacer lo mismo pero en dimensiones más grandes.

Exercise 1

Sea V⊂H V H un subespacio de un espacio de Hilbert H. Sea x∈H x H dado. Encontrar y∈V y V que mejor se aproxime xx. es decir, ∥x−y∥ x y esta minimizada.

Solution

  1. Encontrar una base ortonormal b1…bk b1 … bk para VV
  2. Proyectar xx sobre VV usando y=∑i=1k〈(x,bi)〉⁢bi y i 1 k x bi bi después yy es el punto más cercano en V a x y (x-y) ⊥ V ( ∀v,∀⁢v∈V:〈x−y,v〉=0 v ∀ v V x y v 0

Ejemplo 1

x∈R3 x 3 , V=espacio generado⁢( 1 0 0 )( 0 1 0 ) V espacio generado 1 0 0 0 1 0 , x=( a b c ) x a b c . Por lo tanto, y=∑i=12〈(x,bi)〉⁢bi=a⁢( 1 0 0 )+b⁢( 0 1 0 )=( a b 0 ) y i 1 2 x bi bi a 1 0 0 b 0 1 0 a b 0

Ejemplo 2

V = {espacio de las señales periódicas con frecuancia no mayor que 3⁢ w0 3 w0 }. Dada f(t) periódica, ¿Cúal es la señal en V que mejor se aproxima a f?

  1. { 1T⁢ei⁢ w0 ⁢k⁢t 1 T w0 k t , k = -3, -2, ..., 2, 3} es una ONB para V
  2. g⁢t=1T⁢∑k=-33〈(f⁢t,ei⁢ w0 ⁢k⁢t)〉⁢ei⁢ w0 ⁢k⁢t g t 1 T k -3 3 f t w0 k t w0 k t es la señal más cercana en V para f(t) ⇒ reconstruya f(t) usando solamente 7 términos de su serie de Fourier .

Ejemplo 3

Sea V = { funciones constantes por trozos entre los números enteros}

  1. ONB para V.

bi ={1  if  i−1≤t<i0  otherwise   bi 1 i 1 t i 0 donde {bibi} es una ONB.

¿La mejor aproximación constante por trozos? g⁢t=∑i=−∞∞〈(f, bi )〉⁢ bi g t i f bi bi 〈f, bi 〉=∫−∞∞f⁢t⁢ bi ⁢tdt=∫i−1if⁢tdt f bi t f t bi t t i 1 i f t

Ejemplo 4

Esta demostración explora la aproximación usando una base de Fourier y una base de las ondoletas de Haar. Véase aqui para las instrucciones de como usar el demo.

LabVIEW Example: (run) (source)

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