Find continuous signal given a condition on its samples Let $x(t)$ be band-limited with $B = \omega_m$. Sampling gives us $$x(nT_s) = \begin{cases}
1,  &amp; n = 0 \\
0, &amp; n \not = 0
\end{cases}$$
And $\omega_s = 2\omega_m = \frac{2\pi}{T_s}$. Find signal $x(t)$.
My try: The first problem is about the definition for band-limited signal. It means $X(j\omega) = 0$ for $|\omega|\gt\omega_m$ or $X(j\omega) = 0$ for $|\omega|\ge\omega_m$? The sampling theorem requires that if $X(j\omega) = 0$ for $|\omega|\gt\omega_m$ then $\omega_s \gt 2\omega_m$ to avoid aliasing. So in this case is it possible to find interpolating functions other than $$x(t) = \begin{cases}
\frac{\sin(\omega_mt)}{\omega_mt},  &amp; t \not= 0 \\
1, &amp; t  = 0
\end{cases}$$which I've found using ideal low pass filter? I mean with the given information is $x(t)$ necessarily unique?
The condition
$$x(nT)=\delta[n]\tag{1}$$
is called the Nyquist criterion for zero intersymbol interference (ISI). It is important for the design of transmit pulses in digital communication systems.
Condition $(1)$ can be expressed in the frequency domain as
$$\frac{1}{T}\sum_{k=-\infty}^{\infty}X\left(\omega-\frac{2\pi k}{T}\right)=1\tag{2}$$
where $X(\omega)$ is the Fourier transform of $x(t)$. From $(2)$ we see that the shifted spectra need to add up to a constant. This is only possible if the maximum frequency of $x(t)$ satisfies $\omega_m\ge \pi/T$. For the minimum value $\omega_m=\pi/T$, the shifted spectra don't overlap and, consequently, $x(t)$ must be an ideal low pass signal with a flat spectrum. This is the only solution and it corresponds to the solution you came up with. If $\omega_m&gt;\pi/T$, there are infinitely many solutions to $(2)$. One well-known example are the raised-cosine pulses.