One of the (many) challenging aspects of my Math 305 course (which is roughly a math methods course for scientists) is to help students build intuition for the Fourier transform. Roughly speaking, my goal is for students to not only know how to use Fourier transforms to address problems, but also to gain some intuition for the interplay between “physical space” and “transform space” (also called “k-space” in some physics circles).
(In some sense, what I’m trying to do is to communicate the ideas underlying microlocal analysis without buiding up any of the machinery. Now that I’ve written that, it occurs to me that it could be useful to check out a couple of introductory microlocal analysis books…)
At a first pass, one can look at what it means to be “concentrated” in one space or another. By playing around with functions of the form or characteristic functions, one can pretty quickly develop intuition for the idea that begin “concentrated” in one space roughly corresponds to being “spread out” in another.
Today it occurs to me that it might be very helpful to introduce this same idea when studying cosine/sine Fourier series. A really good exercise would be to plot the magnitude of the coefficients (as a function of frequency) for various functions… and then have students construct functions with various types of frequency distributions. For example, what does a function having equal amounts of the first five frequencies, and no others, look like on physical space? What happens if we toss in equal amounts of frequencies 25-30? Etc.
If this is done at the leve of the Fourier series (and perhaps again at the level of other Fourier-type series, such as Hermite, Legendre, Bessel), then perhaps it will be more natural when we come to Fourier transforms. One can even assign essentially the same problem: Build a function with compact support in frequency space. What does it look like in physical space? What happens if we add a function with frequency support in ? Etc.