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 $z_{1}(x)$ and $z_{2}(x)$. The energy includes not only the stretching of the rubber bands, but also a term proportional to the (positive) area enclosed between them, which is

$\int_{0}^{L}|z_{1}(x) - z_{2}(x)|dx$

Hence my question. So it would be nice to know how this area can be related to the Fourier coefficients of $z_{1}$ and $z_{2}$ or perhaps just to the arc-lengths of the rubber bands. By "statistical correlations" I am referring to the Boltzmann probability distribution with energy equal to the stretching energy plus the area-energy.