Find continuous signal given a condition on its samples
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.