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Teoremas de Plancharel y Parseval

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

Based on: Plancharel and Parseval's Theorems by Justin Romberg

Summary: Este modulo contiene la definición del teorema de Plancharel y del teorema de Parseval con sus demostraciones y ejemplos.

Teorema de Plancharel

Theorem 1: Teorema de Plancharel

El producto interno de dos vectores/señales es el mismo que en ℓ2 ℓ 2 el producto interno de su expansión de coeficientes.

Sea b i b i una base ortonormal para un Espacio de Hilbert H H. x∈H x H , y∈H y H x=∑ i α i ⁢ b i x i α i b i y=∑ i β i ⁢ b i y i β i b i entonces 〈x,y〉 H =∑ i α i ⁢ β i ¯ x y H i α i β i

Ejemplo

Aplicando las Series de Fourier, podemos ir de f⁢t f t a c n c n y de g⁢t g t a d n d n ∫0Tf⁢t⁢g⁢t¯d t =∑ n =−∞∞ c n ⁢ d n ¯ t 0 T f t g t n c n d n el producto interno en el dominio-tiempo = producto interno de los coefientes de Fourier.

Proof

x=∑ i α i ⁢ b i x i α i b i y=∑ j β j ⁢ b j y j β j b j 〈x,y〉 H =〈∑ i α i ⁢ b i ,∑ j β j ⁢ b j 〉=∑ i α i ⁢〈( b i ,∑ j β j ⁢ b j )〉=∑ i α i ⁢∑ j β j ¯⁢〈( b i , b j )〉=∑ i α i ⁢ β i ¯ x y H i α i b i j β j b j i α i b i j β j b j i α i j β j b i b j i α i β i usando las reglas del producto interno.

nota:
〈 b i , b j 〉=0 b i b j 0 cuando i≠j i j y 〈 b i , b j 〉=1 b i b j 1 cuando i=j i j

Si el espacio de Hillbert H tiene un ONB, los productos internos son equivalentes a los productos internos en ℓ2 ℓ 2 .

Todo H con ONB son de alguna manera equivalente a ℓ2 ℓ 2 .

punto de interes:
las secuencias de cuadrados sumables son importantes.

Teorema de Parseval

Theorem 2: Teorema de Parseval

La energía de una señal = suma de los cuadrados de su expansión de coeficientes.

Sea x∈H x H , b i b i ONB

x=∑ i α i ⁢ b i x i α i b i Entonces ∥x∥H2=∑ i | α i |2 H x 2 i α i 2

Proof

Directamente de Plancharel ∥x∥H2= 〈x,x〉 H =∑ i α i ⁢ α i ¯=∑ i | α i |2 H x 2 x x H i α i α i i α i 2

Ejemplo

Series de Fourier 1T⁢ei⁢ w 0 ⁢n⁢t 1 T w 0 n t f⁢t=1T⁢∑ n c n ⁢1T⁢ei⁢ w 0 ⁢n⁢t f t 1 T n c n 1 T w 0 n t ∫0T|f⁢t|2d t =∑ n =−∞∞| c n |2 t 0 T f t 2 n c n 2

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