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Convergencia Uniforme de Secuencias de Funciones.

Module by: Michael Haag, Richard Baraniuk Translated By Fara Meza, Erika JacksonBased 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.

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Convergencia Uniforme de Secuencias de Funciones

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

Definition 1: Convergencia Uniforme
La secuencia gn|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 n≥N n N implica que
| g n t−gt|≤ε g n t g t ε (1)
para todo t∈ℝ 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 t∈ℝ t hay un entero NN que depende de εε y tt tal que ecuación 1 se mantiene si n≥N 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 t∈ℝ t .

Ejemplo 1

∀t,t∈ℝ: g n t=1n t t g n t 1 n Sea ε>0 ε 0 dado. Entonces escoja N=⌈1ε⌉ N 1 ε . Obviamente, ∀n,n≥N:| g n t−0|≤ε n n N g n t 0 ε para toda tt. Así, g n t g n t converge uniformemente a 00.

Ejemplo 2

∀t,t∈ℝ: g n t=tn 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 t→gt 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=1ntift>00ift≤0 g n t 1 n t t 0 0 t 0

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