Let fix $c_1, \ldots, c_n \in \mathbb{C}$ for some $n \geq 1$. I would like to know when does it exist a probability measure $\mathcal{P}$ over the torus $\mathbb{T} = \mathbb{R} / \mathbb{Z}$ such that $\widehat{\mathcal{P}} [k] = c_k$ for $k = 1, \ldots, n$, where $\widehat{\mathcal{P}}[k] = \int_{\mathbb{T}} \mathrm{e}^{2 \mathrm{i} \pi k x} \mathrm{d} \mathcal{P} (x)$.
One necessary condition is that $\lvert c_k \rvert \leq 1$ for any $k$ due to $$|\widehat{\mathcal{P}} [k] | \leq \int_{\mathbb{T}} \mathrm{d}\mathcal{P}(x) = 1.$$
Question: Is the proposed condition sufficient for the existence of $\mathcal{P}$?