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