For a suitably regular continuous function $f \colon S^1 \to \mathbb C$, it is straightforward to see (and well-known) that the zero-th term of its Fourier series $\hat f(n) = \int_0^1 f(t) e^{-2\pi i n t} dt$ represents the average value of the function $f$.
For my purpose, I am interested in the second term $\hat f(2) = \int_0^1 f(t) e^{-4\pi i t} dt$ of the Fourier series. Is anyone aware of any significant meaning of the second term of the Fourier series? In particular, what would $\hat f(2) = 0$ tell about the function $f$ itself, aside from the obvious conclusion that $f$ has no second harmonic?
I think the standard interpretation is: if we set $g(t) = f(t) + f(t+\frac12)$, then the mass of $g(t)$ is somewhat equally distributed in the interval $[0,\frac12]$.
Geometrically, if we think of the domain of $f$ as being in the unit circle in the complex plane, and set $h(z)=f(2z)$, then the values of $h(z)$ on $S^1$ have the origin as their "center of mass".