I have a PDE optimization problem, and a scalar field (which I am optimizing over) is supposed to be nonnegative everywhere in the domain. Since I am working in Fourier space for solving this problem numerically, I need to convert this non-negative constraint into Fourier domain (i.e. constraint on Fourier coefficients).
Assuming the solution is smooth, what is best way to go about it? I really can't think of anything more complicated than just choosing a (large?) grid of points in space, say $\mathbf{x_l}, l=1,2...,M$, and adding M linear constraints of the type:
$$f(\mathbf{x_l})=\sum_{j=1}^N\hat f_\mathbf{j}e^{i2\pi \mathbf{j}.\mathbf{x_l}}\geq 0.$$
I believe the smoothness of $f$ will ensure that this suffices.
Does anyone have any other ideas? Thoughts on this?