Is there a particular transformation that can be done on a function to determine whether it has a lattice structure?

I am dealing with 2D lattices only. Some solutions to a PDE I am interested in consists of functions the minima of which organise themselves in an ordered periodic way: for instance they might arrange in a square lattice, or triangular lattice, or oblique with a given angle. The lattices might even be non-point lattices (but still periodic): example honeycomb or kagome lattice.

I was wondering whether it exists,given a periodic lattice, a transformation that can be done on the lattice which allows for the detection of the structure of the lattice or helps its classification. More specifically I was thinking whether one has in mind a clever way to use Fourier transforms or something of the sort to obtain a parameter which describes such lattices.

Otherwise any relevant reference on the topic? There must be someone studying these systems even though I could not find much in the literature.

• There are four tags including "lattice" in their name. Are you sure your question is about "lattice-orders"? Perhaps you would be more likely to get an answer with some other tag... – amrsa Dec 14 '16 at 11:07
• @amrsa it is not about any of the pre existing tags but unfortunately I cannot create a new one. Anyways I really wouldn't know what other tag to include. Maybe crystallography; apart from that I really wouldn't know. That's probably also why I struggled to find something in the literature. – semola Dec 15 '16 at 14:24

I think you're defining "lattice" as "isomorphic to $\mathbb Z ^ n$ (or particularly $\mathbb Z ^ 2$).