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.
Edit: Specifics on the Boltzmann probability distribution, more motivation.
The state of the system is the pair of functions $z_{1}(x)$ and $z_{2}(x)$ describing the deviation of the two rubber bands from the x axis. Let's say it's the set of pairs of functions defined on [0,L] and that these functions are identified with a finite number of Fourier coefficients – I am a physicist and would like to avoid nasty functions or mathematically honest discussions of path integrals.
The probability of occurrence of a state (z_{1}, z_{2}) is (before normalization)
$\exp(-\beta E\left[z_{1},z_{2}\right])$
where $E\left[z_{1},z_{2}\right]$ is the energy of the system, which in this case is the functional
$E\left[z_{1},z_{2}\right] = \frac{t}{2}\int_{0}^{L}\left[(\frac{dz_{1}}{dx})^{2}+(\frac{dz_{2}}{dx})^{2}\right]dx + \kappa \int_{0}^{L}|z_{1}(x)-z_{2}(x)|dx$
Where the tension t and the "surface tension" $\kappa$ are just numbers; set them equal to 1 if you wish. The first integral is the energy cost of stretching the rubber bands (in a linearized regime) and the second is the strange term proportional to the area enclosed between them. Without the second term, it is easy to diagonalize this functional in terms of the Fourier series of the two functions. That is why I was interested in writing the second term in terms of Fourier coefficients. That may be too much to ask, but perhaps it is still possible to calculate some quantities such as the statistical average of $z_1(x)^{2}$ – that is the kind of thing I ultimately want to know.
I realize this is an unnatural-looking problem, so I will just mention that it's not really about rubber bands, but rather about fluctuating interfaces which occur in lipid bilayers with coexisting phases. There are two phase boundaries (one for each monolayer) with their respective "tensions" but there is also a term proportional to the area between them.
Best Answer
This sounds difficult. For instance, a long-standing open problem of Littlewood used to be this. Let A be a set of integers of size n, and let f be the characteristic function of A. How small (up to a constant) can the sum of the absolute values of the Fourier coefficients of f be? The conjecture was that the smallest was $C\log n$, which is what you get when A is an arithmetic progression. This conjecture is now known to be correct, but plenty of closely related questions are still open. So at least sometimes the relationship between the $L_1$ norm of a function and the Fourier transform of that function is quite hard to understand.