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Convergencia Uniforme de Secuencias de Funciones.

Module by: Michael Haag, Richard Baraniuk. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Uniform Convergence of Function Sequences by Michael Haag, Richard Baraniuk

Summary: Otra forma de convergencia, convergencia uniforme es definida y descrita en este modulo. También se muestra su relación con la convergencia puntual.

Convergencia Uniforme de Secuencias de Funciones

para esta discusión, solo consideraremos las funciones con g n g n donde RR

Definition 1: Convergencia Uniforme
La secuencia g n | n =1 n 1 g n converge uniformemente a la funció gg si para cada ε>0 ε 0 existe un entero NN tal que nN n N implica que
| g n tgt|ε g n t g t ε
(1)
para todo tR t .
Obviamente toda secuencia uniformemente continua es de convergencia puntual . La diferencia entre convergencia puntual y uniformemente continua es esta: Si g n g n converge puntualmente a gg, entonces para todo ε>0 ε 0 y para toda tR t hay un entero NN que depende de εε y tt tal que ecuación 1 se mantiene si nN n N . Si g n g n converge uniformemente a gg, es posible que para cada ε>0 ε 0 enocntrar un entero NN que será par todo tR t .

Ejemplo 1

g n t=1n  ,   tR    t t g n t 1 n Sea ε>0 ε 0 dado. Entonces escoja N=1ε N 1 ε . Obviamente, | g n t0|ε  ,   nN    n n N g n t 0 ε para toda tt. Así, g n t g n t converge uniformemente a 00.

Ejemplo 2

g n t=tn  ,   tR    t t g n t t n Obviamente para cualquier ε>0 ε 0 no podemos encontrar una función sencilla g n t g n t para la cual la ecuación 1 se mantiene con gt=0 g t 0 para todo tt. Así g n g n no es convergente uniformemente. Sin embargo tenemos: g n tgt puntual g n t g t puntual

conclusión:

La convergencia uniforme siempre implica convergencia puntual, pero la cpnvergencia puntual no necesariamente garantiza la convergencia uniforme.

Problems

Pruebe rigurosamente si las siguientes funciones convergen puntualmente o uniformemente, o ambas.

  1. g n t=sintn g n t t n
  2. g n t=etn g n t t n
  3. g n t={1nt  if  t>00  if  t0 g n t 1 n t t 0 0 t 0

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