College Math Teaching

August 7, 2014

Engineers need to know this stuff part II

This is a 50 minute lecture in a engineering class; one can easily see the mathematical demands put on the students. Many of the seemingly abstract facts from calculus (differentiability, continuity, convergence of a sequence of functions) are heavily used. Of particular interest to me is the remarks from 45 to 50 minutes into the video:

Here is what is going on: if we have a sequence of functions $f_n$ defined on some interval $[a,b]$ and if $f$ is defined on $[a,b]$, $lim_{n \rightarrow \infty} \int^b_a (f_n(x) - f(x))^2 dx =0$ then we say that $f_n \rightarrow f$ “in mean” (or “in the $L^2$ norm”). Basically, as $n$ grows, the area between the graphs of $f_n$ and $f$ gets arbitrarily small.

However this does NOT mean that $f_n$ converges to $f$ point wise!

If that seems strange: remember that the distance between the graphs can say fixed over a set of decreasing measure.

Here is an example that illustrates this: consider the intervals $[0, \frac{1}{2}], [\frac{1}{2}, \frac{5}{6}], [\frac{3}{4}, 1], [\frac{11}{20}, \frac{3}{4}],...$ The intervals have length $\frac{1}{2}, \frac{1}{3}, \frac{1}{4},...$ and start by moving left to right on $[0,1]$ and then moving right to left and so on. They “dance” on [0,1]. Let $f_n$ the the function that is 1 on the interval and 0 off of it. Then clearly $lim_{n \rightarrow \infty} \int^b_a (f_n(x) - 0)^2 dx =0$ as the interval over which we are integrating is shrinking to zero, but this sequence of functions doesn’t converge point wise ANYWHERE on $[0,1]$. Of course, a subsequence of functions converges pointwise.