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Uniform Convergence of Function Sequences

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

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 RR

Definition 1: Uniform Convergence
The sequence g n | 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 nN n N implies
| g n tgt|ε g n t g t ε
(1)
for all tR 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 tR t there is an integer NN depending on εε and tt such that Equation 1 holds if nN 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 tR t .

Example 1

t ,tR: 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 ,nN:| g n t0|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 ,tR: 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 tgt pointwise g n t g t pointwise

Conclusion:

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

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