Consider the signal $g(t)=\cos(2\pi \lambda t+\phi)$ that is sampled with a frequency $\tau$. Let $g_k$ denote the values of $g$ at the times $t_k=\frac{k}{\tau}$, $k \in \mathbb{N}$.
(a) Show that there exists a frequency $f\in[-\frac{\tau}{2},\frac{\tau}{2})$ such that the signal $h(t)=\cos(2\pi f t+\phi)$ gives the same samples as $g$ at the times $t_k$.
(b) Deduce from this result a necessary condition (involving $\lambda$ and $\tau$) for $g$ to be completely determined by its samples.
I think that the issue is linked to the Nyquist–Shannon sampling theorem, in the sense that given a sample we can get the signal depending on whether certain conditions on sampling and the signal are satisfied. I don't know if the relationship is bijective, or how to use this theorem in question, if it is suitable to the case.
In (a) the difficulty I'm having is to show that a suitable $f$ is $[-\frac{\tau}{2}, \frac{\tau}{2})$.
In (b) I think that it is necessary to write the Nyquist frequency in terms of $\lambda$ and $\tau$ so that for all frequency below this the theorem guarantees that it is possible to recover the signal.
This notation apparently different is used by the book of Gasquet/Witomski of Fourier Analysis. I have not written exercise in another notation to leave as it appears in the book.