Uniform Convergence of Function Sequences 2.4 2002/10/11 2005/06/24 20:33:26.002 GMT-5 Michael Haag mjhaag@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Richard G. Baraniuk richb@rice.edu converge convergence converges function sequences sequences uniform uniform convergence 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 where Uniform Convergence The sequence n 1 g n converges uniformly to function g if for every ε 0 there is an integer N such that n N implies g n t g t ε for all t . Obviously every uniformly convergent sequence is pointwise convergent. The difference between pointwise and uniform convergence is this: If g n converges pointwise to g, then for every ε 0 and for every t there is an integer N depending on ε and t such that holds if n N . If g n converges uniformly to g, it is possible for each ε 0 to find one integer N that will do for all t . t t g n t 1 n Let ε 0 be given. Then choose N 1 ε . Obviously, n n N g n t 0 ε for all t. Thus, g n t converges uniformly to 0. t t g n t t n Obviously for any ε 0 we cannot find a single function g n t for which holds with g t 0 for all t. Thus g n is not uniformly convergent. However we do have: g n t g t pointwise 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. g n t t n g n t t n g n t 1 n t t 0 0 t 0