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.


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