Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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.

share|improve this answer
    
I might need to clarify the problem a little bit. When I say the function is band-limited with m lower frequency components, I'm not saying that it is with 'some' m, but a 'fixed' m, i.e. 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. Therefore, the condition I'm seeking should contain (and depend on) m. –  David Feb 24 '11 at 4:46
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.