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Este módulo presenta dos tipos comunes de convergencia, puntual y norma, discutiremos sus propiedades, diferencias y relaciones entre ellos.

Convergencia de vectores

Discutiremos la convergencia puntual y de la norma de vectores. También existen otros tipos de convergencia y uno en particular, la convergencia uniforme , también puede ser estudiada. Para esta discusión, asumiremos que los vectores pertenecen a un espacio de vector normado .

Convergencia puntual

Una secuencia n 1 g n converge puntualmente al límite g si cada elemento de g n converge al elemento correspondiente en g . A continuación hay unos ejemplos para tratar de ilustrar esta idea.

g n g n 1 g n 2 1 1 n 2 1 n Primero encontramos los siguiente limites para nuestras dos g n 's: n g n 1 1 n g n 2 2 Después tenemos el siguiente, n g n g puntual, donde g 1 2 .

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t t g n t t n Como se hizo anteriormente, primero examinamos el límite n g n t 0 n t 0 n 0 donde t 0 . Por lo tanto n g n g puntualmente donde g t 0 para toda t .

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Norma de convergencia

La secuencia n 1 g n converge a g en la norma si n g n g 0 . Aqui ˙ es la norma del espacio vectorial correspondiente de g n 's. Intuitivamente esto significa que la distancia entre los vectores g n y g decrese a 0 .

g n 1 1 n 2 1 n Sea g 1 2

g n g 1 1 n 1 2 2 1 n 1 2 1 n 2 1 n 2 2 n
Asi n g n g 0 , Por lo tanto, g n g en la norma.

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g n t t n 0 t 1 0 Sea g t 0 para todo t .

g n t g t t 1 0 t 2 n 2 n 0 1 t 3 3 n 2 1 3 n 2
Asi n g n t g t 0 Por lo tanto, g n t g t en la norma.

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Puntual vs.norma de convergencia

Para m , la convergencia puntual y la norma de convergencia es equivalente.

Puntual ⇒ norma

g n i g i Asumiendo lo anterior, entonces g n g 2 i m 1 g n i g i 2 Así,

n g n g 2 n i m 1 g n i g i 2 i m 1 n g n i g i 2 0

Norma ⇒ puntual

g n g 0

n i m 1 g n i g i 2 i m 1 n g n i g i 2 0
Ya que cada término es mayor o igual a cero, todos los términos' m ' deben ser cero. Así, n g n i g i 2 0 para todo i . Por lo tanto, g n g puntual

En un espacio de dimensión finita el teorema anterior ya no es cierto. Probaremos esto con contraejemplos mostrados a continuación.

Contra ejemplos

Puntual ⇒ norma

Dada la siguiente función: g n t n 0 t 1 n 0 Entonces n g n t 0 Esto significa que, g n t g t pointwise donde para todo t g t 0 .

Ahora,

g n 2 t g n t 2 t 1 n 0 n 2 n
Ya que la norma de la función se eleva, no puede converger a cualquier función con norma finita.

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Norma ⇒ puntual

Dada la siguiente función: g n t 1 0 t 1 n 0 si n es par g n t -1 0 t 1 n 0 si n es impar Entonces, g n g t 1 n 0 1 1 n 0 donde g t 0 para todo t . Entonces, g n g en la norma Sin embargo, en t 0 , g n t oscila entre -1 y 1, Y por lo tanto es no convergente. Así, g n t no tiene convergencia puntual.

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Problemas

Pruebe si las siguientes secuencias tienen convergencia puntual, norma de convergencia, o ambas se mantienen en sus limites.

  • g n t 1 n t 0 t 0 t 0
  • g n t n t t 0 0 t 0

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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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