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Espacios de Hilbert comunes

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

Based on: Common Hilbert Spaces by Roy Ha, Justin Romberg

Summary: Este modulo nos dará una descripción de los espacios de Hilbert mas comunes y de sus propiedades básicas.

Espacios de Hilbert comunes

A continuación veremos los cuatro Espacios de Hilbert mas comunes con los que usted tendrá que tratar para la discusión y manipulación de señales y sistemas:

Rn n (escalares reales ) y Cn n (escalares complejos), también llamado ℓ 2 ⁢ 0 n−1 ℓ 2 0 n 1

x= x 0 x 1 … x n - 1 x x 0 x 1 … x n - 1 Es una lista de números (secuencia finita). El producto interno para nuestros dos espacios son las siguientes:

  • Producto interno Rn n :
    〈x,y〉=yT⁢x=∑i=0n−1 x i ⁢ y i x y y x i 0 n 1 x i y i
    (1)
  • Producto interno Cn n :
    〈x,y〉=yT¯⁢x=∑i=0n−1 x i ⁢ y i ¯ x y y x i 0 n 1 x i y i
    (2)

Modelo para: Señal de tiempo discreto en el intervalo 0 n−1 0 n 1 o Señal Periódica (con periodo nn) de tiempo discreto. x 0 x 1 … x n - 1 x 0 x 1 … x n - 1

Figura 1
Figura 1 (fig1.png)

f∈ L 2 ⁢ a b f L 2 a b es una función de energía finita en a b a b

Inner Product

〈f,g〉=∫abf⁢t⁢g⁢t¯dt f g t a b f t g t
(3)
Modelo para: Señal de tiempo continuo en el intervalo a b a b o Señal Periódica (con periodo T=b−a T b a ) de tiempo continuo

x∈ ℓ 2 ⁢Z x ℓ 2 es una secuencia infinita de números que son cuadrados sumables

Producto interno

〈x,y〉=∑i=−∞∞x⁢i⁢y⁢i¯ x y i x i y i
(4)
Modelo para: Señal no-periódica de tiempo discreto

f∈ L 2 ⁢R f L 2 es una una función de energía finita en todo R.

Producto interno

〈f,g〉=∫−∞∞f⁢t⁢g⁢t¯dt f g t f t g t
(5)
Modelo para: Señal no-periódica de tiempo continuo

Análisis de Fourier Asociado

Cada uno de estos cuatro espacios de Hilbert tiene un análisis de Fourier asociado con el.

  • L 2 ⁢ a b L 2 a b → Series de Fourier
  • ℓ 2 ⁢ 0 n−1 ℓ 2 0 n 1 → Transformada Discreta de Fourier
  • L 2 ⁢R L 2 → Transformada de Fourier
  • ℓ 2 ⁢Z ℓ 2 → Transformada Discreta de Fourier en Tiempo
Pero los cuatros están basados en el mismo principio (Espacio de Hilbert).

Nota Importante:

no todos los espacios normalizados son espacios de Hilbert
Por ejemplo: L 1 ( ℝ ) L 1 ( ℝ ) , ∥f∥1=∫|f⁢t|dt 1 f t f t . trate como usted pueda, de encontrar el producto interno que induce esta norma, es decir...tal que: 〈·,·〉 · · such that
〈f,f〉=∫|f⁢t|2dt2=∥f∥12 f f t f t 2 2 1 f 2
(6)
De echo, para todo el espacio L p ⁢R L p , L 2 ⁢R L 2 es el único que es un espacio de Hilbert.

Figura 2
Figura 2 (fig3.png)

Los espacios de Hilbert son en gran medida los más agradables, si se usa o estudia la expansión de bases ortonormales entonces usted empezara a ver por que esto es cierto.

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