Take the 2-minute tour ×  I would like to express the integral
of the absolute value of a real-valued function $f$ (over a
finite interval) in terms of the Fourier coefficients of $f$.
Failing that, I would like to know of any constraints or
statistical correlations (in a sense explained in the
motivation) relating these quantities.  Motivation: This comes
from a biophysics application, but is perhaps best explained as
follows. If a rubber band of tension $t$ is stretched along the
$x$ axis from $0$ to $L$, then it is easy to calculate the
thermal fluctuations of its arc-length by letting $z(x)$ be the
(small) deviation from the $x$ axis, and then writing the energy
(arc-length times tension) in terms of the Fourier coefficients
of $z(x)$. The Boltzmann weight turns out to be a Gaussian since
in the limit of small deviations the arc-length becomes a sum of
squares of the Fourier coefficients. My problem is more
complicated: We have two rubber bands stretched over the same
interval, with deviations $z_{1}(x)$