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Uniform Convergence of Function Sequences

Module by: Michael Haag, Richard Baraniuk. E-mail the authors

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

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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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