Let $\hat{\mu}:\mathbb{Z}\rightarrow\mathbb{C}$ be an arbitrary function. Then, letting $\mathbb{C}\left[e^{2\pi it},e^{-2\pi it}\right]$ denote the vector space space of all trigonometric polynomials (linear combinations of the form $\sum_{n=-N}^{N}a_{n}e^{2n\pi it}$), we can define a linear functional $d\mu:\mathbb{C}\left[e^{2\pi it},e^{-2\pi it}\right]\rightarrow\mathbb{C}$ by way of the formula:$$\int_{0}^{1}e^{-2n\pi it}d\mu\left(t\right)\overset{\textrm{def}}{=}\hat{\mu}\left(n\right),\textrm{ }\forall n\in\mathbb{Z}$$ which we then extend by linearity. My question is: are there any useful criteria on $\hat{\mu}$ to determine whether or not the $d\mu$ thus defined is in fact a finite Borel measure on $\mathbb{R}/\mathbb{Z}$? In particular, I would like to know if there are criteria that are easier to check than the following:
It is well-known that linear combinations of the form $\sum_{n=-N}^{N}a_{n}e^{2n\pi it}$ are dense in the Banach space $C\left(\mathbb{R}/\mathbb{Z}\right)$ of all continuous, $1$-periodic complex-valued functions on the unit interval, with respect to that space's norm (supremum). By standard functional analysis, it is known that every continuous linear functional on $C\left(\mathbb{R}/\mathbb{Z}\right)$ is a finite Borel measure on $\mathbb{R}/\mathbb{Z}$; as such, in order for $d\mu$ to be finite, it is sufficient that:$$\lim_{N\rightarrow\infty}\int_{0}^{1}\left(\sum_{n=-N}^{N}a_{n}e^{-2n\pi it}\right)d\mu\left(t\right)=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N}a_{n}\hat{\mu}\left(n\right)$$ converges to a finite value whenever:$$\sum_{n=-N}^{N}a_{n}e^{-2n\pi it}$$ converges in supremum norm to a function in $C\left(\mathbb{R}/\mathbb{Z}\right)$ as $N\rightarrow\infty$. The reason I say this sufficient condition is “not easy to check” because, for example, you can't assume that the $a_{n}$s are in $\ell^{1}\left(\mathbb{Z}\right)$—in that case, $\sum_{n=-\infty}^{\infty}a_{n}e^{-2n\pi it}$ would be an element of the Wiener algebra on $\mathbb{R}/\mathbb{Z}$, and it is known that there are continuous functions on $\mathbb{R}/\mathbb{Z}$ which are not in the Wiener algebra.
The Fourier-Stieltjes algebra of a discrete group $G$, usually denoted by $B(G)$, was defined by P. Eymard in 1964 to be the linear span, within the space of complex valued functions on $G$, of the positive type functions, namely the functions $f:G\to {\mathbb C}$, such that $$ \sum_{i,j=1}^nc_i\overline {c_j}f(g_j^{-1} g_i)\geq 0, $$ whenever $g_1, g_2, \ldots , g_n\in G$, and $c_1, c_2, \ldots , c_n\in {\mathbb C}$.
As the name suggests, $B(G)$ is indeed an algebra under pointwise multiplicaion of functions.
In the case of an abelian group $G$, such as the one of interest for the OP, the Fourier-Stieltjes algebra is precisely formed by the Fourier transforms of the finite measures on the dual group $\widehat G$.
Summarizing, a function $\hat \mu :{\mathbb Z}\to {\mathbb C}$ satisfies the condition mentioned in the question if and only if it is a linear combination of positive type functions on ${\mathbb Z}$.