Is there any convincing sense in which the standard trigonometric basis for the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is optimal among all the orthonormal bases?

(If this question strikes you as overly subjective, replace "convincing" by "described in the existing mathematical literature".)

Here's an example of the kind of thing that might be true (but please don't confuse this concrete question with the deliberately vague question that I'm really interested in!).

Consider the sequence of functions $\frac{1}{\sqrt{2}}$, $\cos t$, $\sin t$, $\cos 2t$, $\sin 2t$, $\dots$. This is an orthonormal basis for $V$ with respect to the inner product $\langle f,g \rangle = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t)g(t) \: dt$. The functions have respective total variation 0,4,4,8,8,$\dots$. I wishfully conjecture (on the basis of no evidence whatsoever) that for any finite set $\{f_1,\dots,f_n\}$ of orthonormal elements of $V$, the total variation of $f_i$, summed over $i$ going from 1 to $n$, is at least $4 \lfloor n^2/4 \rfloor$ (which is the sum of the first $n$ terms of the sequence $0,4,4,8,8,\dots$). In this sense, the standard Fourier basis is a "least wiggly" basis.

charactersgive the corresponding standard orthonormal basis. – Todd Trimble♦ Jul 24 '13 at 1:57