Question

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.

Answer 1

In the continuous case, assuming the function $f(x_1 , x_2 )$ has a Fourier transform of compact support, by the Paley-Wiener theorem it can be extended to a holomorphic function $f(z_1 , z_2 )$ on $\mathbb{C}^2$. I think this implies that its level sets are real-analytic curves in $\mathbb{R}^2$; so one necessary condition for a curve to be the level-set of $f(x_1 , x_2 )$ is that it's real-analytic (for example, a polygon cannot be a level-set). In the discrete case, you may be able to obtain bounds on the derivatives (curvature) of a level-set in terms of the total power contained in the finitely-many Fourier components.