ChapterZero

We interrupt this regularly scheduled nothing to bring you a new posting:

Have you ever heard someone mention off-handedly that 2π periodic functions can also be viewed as functions defined on the circle, and wondered what in the world… is so obvious about that fact, that they don’t bother to explain themselves… and wondered if that meant you were missing the obvious?

Well, the answer is yes. You were missing the obvious: let x be the angle subtended in the circle, then if f is a 2π periodic function, f(x) represents the function at the location x on the circumference of the circle. Isn’t that neat?

That raises some other ideas in my mind, probably not amounting to anything, but I’ll mention them anyway. What about q-π periodic functions, where q is a real number? It seems like you could do the same thing, and then define an isomorphism between q-π periodic functions and 2-π periodic functions. Actually, that’s obvious even without the circle idea: just use the same abscissa scaling that we use to give trig functions arbitrary periods.

Question: why do we use complex exponentials (and by extension, trig polynomials) as the orthonormal set for determining Fourier series. Other than the historical reasons, ease of determining the general coefficients (you only have to do two or three integrals, at most), and the relatively straight forward physical interpretation it lends itself to? I can’t think of any other closed orthogonal set offhand that it wouldn’t be a pain to orthonomalize, and even more of a pain to determine *all* the Fourier coefficients with respect to for a given function… but that doesn’t mean that they don’t exist.