In a practical application problem I encountered such a question: Given a subset of a N*N Cartesian grid, how to determine if it is a sublevel-set of a band-limited (discrete) function? Here band-limited function means, say, it has only m lower frequency components.

Of course this question has a continuous form: What's the property that a curve must satisfy for it being the levelset of a periodic band-limited function with m lower frequency components?

In the above questions when I say the function is band-limited with m lower frequency components, I'm not talking about the set of all function with finite frequency components, but the set of all function with less than m frequency components where m is a given integer. Therefore, the condition I'm seeking should contain (and depend on) m.

The answer to any of these two forms, or related references would be helpful. Thanks a lot.