Can I relate the L1 norm of a function to its Fourier expansion?

Question

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.

Answer 1

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.

Answer 2

This answer really has to do with the physics of it: are you sure about the area energy term?

Let me simplify a little bit: Consider a case where $z_1(x)=0$ and $z_2(x)=w(x)$. According to your formula I should get a term proportional to $w'(x)^2$ for the kinetic term which is typical and no one will object to you for that. You potential energy would then be proportional to $\int_0 ^L |w(x)| dx$. That does raise an alarm: nonlinear problem.

Unless you really have nonlinear physics going on (which is likely, see below) you should have had $w(x)^2$. That would bring back the $L_2$ norm and everything will be very simple. Just transform $z_2(x)=z_1(x)+w(x)$, get rid of $z_2$ and you will get two uncoupled (continuum of) modes, one (corresponding to $z_1$) is a free particle and the other a Harmonic oscillator. Your Boltzmann statistic will determine how these modes are filled up as you know.

If you think of your rubber band as a collection of springs and masses, $w(x)^2$ is the actual term but collections of springs and masses hardly exist outside textbooks and problem sets. As a physicist you know: molecules interact. The actual expansion (if you are still interested in writing an effective field theory) will involve (interacting) terms of the type $w(x)^2+O[w(x)^4]$. Quantitative results in this model might involve renormalization. See the wikipedia page on quartic interaction that even describes how you should quantize it (Fourier transform). My field theory is rusty so you might already know more than I do.

What if $|w(x)|$ is what you have: If $|w(x)|$ is not too small or too large, just approximate it with $[w(x)]^2$ to linearize the problem. People might point out that this is not a mathematically good approximation but it will physically make sense: very small stretchings are not physically possible and very large stretchings would be ruled out by the Boltzmann statistics as they would correspond to exponentially rare high energy modes. So I would just introduce a factor so that $w(x)$ and $w(x)^2$ coincide where $\kappa w(x)^2 L\approx kT$.

There are a famous nonlinear equations that come cheerfully close to your problem but miss it. Example: Sine-Gordon Equation where instead of $|w(x)|$ you would have $1-\cos [w(x)]$. Actually more up to the point would be the Sinh-Gordon... which reminds me of the Toda field theory which describes a Toda lattice. A Toda lattice is a nonlinear set of coupled equations that describe a set of nonlinearly coupled particles. If you are working in the liquid state, I doubt that they would be relevant. $w(x)^2$ should be good enough but the dissipative terms will be more troublesome.

Edit: Couldn't resist the pun. Seems that the relevant equation is a "SIGN-Gordon equation": $$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$ Not sure if it is really simple, messy, or plain difficult to solve in your case. An option is to try solve it like a wave equation with a sign changing external force term and then parametrize the solutions based on their energy and apply the Boltzmann statistics. Forgo the Hamiltonian altogether. I think it will be messy.

Another Edit: as the commenters mentioned before, the solution to the absolute value potential Schrodinger eqn. involves Airy functions. So you could in principle use the Airy function eigenstates to find the time-independent modes and fix their energy. This could quantize your problem. The solution in terms of the Airy functions can be found for example here, [PRL, 94, 176805 (2005)].

Yet Another Update: Apparently you have the "Signum-Gordon" equation. Thanks to this answer.