3
$\begingroup$

The classical Shannon sampling theorem states that a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $\hat f$ refers to the Fourier transform of $f$).

My question is as follows: Suppose that $f$ is not bandlimited but that we have an estimate of the form

$$ |\hat f (\omega)|\le C \exp(-|\omega|^2). $$

Can such $f$ still be determined by its samples at the integers?

Edit: The answer to the question as stated is `NO', by the example of Jean Duchon. But is the following true: Suppose that $\hat f$ satisfies the above growth estimate. Can $f$ be uniquely determined by its samples $(f(\alpha i))_{i\in\mathbb{Z}}$ for all $\alpha <1$?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ I don't understand the update - one can write each real number as $\alpha i$ with $i\in\mathbb Z$ and $0<\alpha<1$. Do you mean for some $\alpha<1$? $\endgroup$
    – Sebastian Goette
    Mar 3, 2016 at 16:04
  • $\begingroup$ Why not? the lattice $\mathbb{Z}$ is not dense enough for sampling these functions, so the question is whether the denser lattices $\alpha \mathbb{Z}$, $\alpha <1$ are sampling sets. $\endgroup$
    – conan
    Mar 3, 2016 at 16:07
  • $\begingroup$ The edited question is unclear. Every number can be written as $\alpha i, i\in Z, 0<\alpha<1$. $\endgroup$
    – Alexandre Eremenko
    Mar 3, 2016 at 20:54
  • $\begingroup$ why don't you try to understand how works the sampling theorem, i.e. how sampling in the time domain corresponds to periodization in the frequency domain ? this is linked to the Poisson summation formula which says in reality that (in the sense of distributions) the Dirac comb $\sum_n \delta(x-n)$ ) is its own Fourier transform. hence the answer is no, the perioziation operator is invertible only for those functions compactly supported on less than one period. $\endgroup$
    – reuns
    Mar 4, 2016 at 1:07

2 Answers 2

Reset to default
3
$\begingroup$

No as well to the edited question (if I understood it...): $f(x)=e^{-cx^2}\sin(\pi x/\alpha)$

Improve this answer
$\endgroup$
3
$\begingroup$

No. Example $f(x)=e^{-cx^2}\sin(\pi x)$, chosing $c$ in the proper range...

Improve this answer
$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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