College Math Teaching

April 17, 2012

Pointwise versus Uniform convergence of sequences of continuous functions: Part II

Filed under: analysis, calculus, complex variables, uniform convergence — collegemathteaching @ 12:48 am

In my complex analysis class I was grading a problem of the following type:
given K \subset C where K is compact and given a sequence of continuous functions f_n which converges to 0 pointwise on K and if |f_1(z)| > |f_2(z)|>...|f_k(z)|... for all z \in K show that the convergence is uniform.

In a previous post, I talked about why it was important that each f_k(z) be continuous.

Now what about the |f_1(z)| > |f_2(z)|>...|f_k(z)|... hypothesis? Can it be dispensed with?

Answer: well, no.

Let’s look at an example in real variables:

Let f_n(x) = sin(\frac{e \pi}{2}e^{-nx})sin(\frac{n \pi}{2} x) with x \in [0,1] . f_n(0) = 0 for all n . To see that f_n converges to zero pointwise, note that lim_{n \rightarrow \infty}e^{-nx} = 0 for all x > 0 , hence lim_{n \rightarrow \infty}sin(\frac{e \pi}{2}e^{-nx}) = 0 which implies that f_n \rightarrow 0 by the squeeze theorem. But f_n does not converge to 0 uniformly as for t = \frac{1}{n} we have f_n(t) = 1

Here is a graph of the functions for n = 5, 10, 20, 40

1 Comment »

  1. […] the case of pointwise convergence of functions (studied earlier in this blog here and here. […]

    Pingback by Convergence of functions and nets (from advanced calculus) | College Math Teaching — January 17, 2015 @ 4:57 am

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