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Uniform Convergence of Function Sequences

Module by: Michael Haag, Richard Baraniuk

Summary: Another form of convergence, uniform convergence, is defined and described in this module. Also, its relationship to pointwise convergence is also shown.

Uniform Convergence of Function Sequences

For this discussion, we will only consider functions with g n g n where ℝ→ℝ

Definition 1: Uniform Convergence
The sequence gn|n=1∞ n 1 g n converges uniformly to function gg if for every ε>0 ε 0 there is an integer NN such that n≥N n N implies
| g n t-gt|≤ε g n t g t ε (1)
for all t∈ℝ t .
Obviously every uniformly convergent sequence is pointwise convergent. The difference between pointwise and uniform convergence is this: If g n g n converges pointwise to gg, then for every ε>0 ε 0 and for every t∈ℝ t there is an integer NN depending on εε and tt such that Equation 1 holds if n≥N n N . If g n g n converges uniformly to gg, it is possible for each ε>0 ε 0 to find one integer NN that will do for all t∈ℝ t .

Example 1

∀t,t∈ℝ: g n t=1n t t g n t 1 n Let ε>0 ε 0 be given. Then choose N=⌈1ε⌉ N 1 ε . Obviously, ∀n,n≥N:| g n t-0|≤ε n n N g n t 0 ε for all tt. Thus, g n t g n t converges uniformly to 00.

Example 2

∀t,t∈ℝ: g n t=tn t t g n t t n Obviously for any ε>0 ε 0 we cannot find a single function g n t g n t for which Equation 1 holds with gt=0 g t 0 for all tt. Thus g n g n is not uniformly convergent. However we do have: g n t→gt pointwise g n t g t pointwise

conclusion:

Uniform convergence always implies pointwise convergence, but pointwise convergence does not guarantee uniform convergence.

Problems

Rigorously prove if the following functions converge pointwise, uniformly, or both.

  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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