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Band-limited function, Shannon-Nyquist sampling distance

  1. Oct 16, 2015 #1
    1. The problem statement, all variables and given/known data
    If ##\delta## and ##\Omega## are two numbers with ##0 < \frac{\pi}{\Omega} < \delta## find a function ##f\in L^2(\mathbf R )## such that ##\hat f(\omega)=0## for ##|\omega|> \Omega## and ##f(n\delta) = 0## for ##n \in \mathbf Z##, but ##f\ne 0## as an element of ##L^2(\mathbf R )##. This shows that the Shannon-Nyquist sampling distance ##\Delta t = \frac{\pi}{\omega_{\max}}## is the largest possible.

    2. Relevant equations
    Fourier transform
    ##\frac{\sin ax}{x} \to \pi \mathscr{X}_a(\xi ) =
    \begin{cases}
    \pi \; \; |\xi | < a \\
    0 \; \; |\xi | > a
    \end{cases}##

    3. The attempt at a solution
    I Choose ##f(t) = \frac{\sin \pi \delta t}{t}## which seems to satisfy all the criteria since
    ##F.T.(\frac{\sin \pi \delta t}{t}) = \pi \mathscr{x}_{\pi /\delta } < \pi \mathscr{x}_{\pi /\Omega }##
    Is this what I was supposed to do in the exercise? There's no answer so I'm not sure I proved everything I should. What does it mean that ##f\ne 0## as an element of ##L^2(\mathbf R )##? That it's not the zero-function?
     
    Last edited: Oct 16, 2015
  2. jcsd
  3. Oct 16, 2015 #2
    I assume you meant "##\delta## and ##\Omega## are two numbers...". Shouldn't it be ##f(t) \propto \frac{\sin \pi t/\delta}{t}## ? Even still it fails for ##n=0##. But this could still be the answer they want. Yes, ##f\neq 0 ## means it's not the zero function (which is a trivial solution).
     
  4. Oct 16, 2015 #3
    Yes should've been ##\delta##. Your version of ##f(t)## is of cause equally correct (and more general). Since they only ask me to find a solution I didn't bother with the constant. I think I take your post as confirmation that at least I probably got what the question asked for. I guess there may be a solution with ##f(0)=0## or atl where ##\lim_{x\to 0} f(x) =0## but I think they probably meant except that point.
    I guess I could also choose some function that is zero past a point say ##|x| > a## and get other solutions than trigonometric functions and have convergence even without a ##t## in the denominator but I feel quite happy as it is!
     
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