Background Given a (translation bounded) positive definite measure $\gamma$ lets say on $\mathbb R^d$, its Fourier transform as a tempered distribution is a positive measure $\widehat{\gamma}$.
I am interested in the Lesbegue decomposition
$$\widehat{\gamma}=\widehat{\gamma}_{pp}+\widehat{\gamma}_{ac}+\widehat{\gamma}_{sc}$$
at the level of $\gamma$. The decomposition $\widehat{\gamma}=\widehat{\gamma}_{pp}+\widehat{\gamma}_{c}$ is well understood, so the problem reduces simply to understanding the decomposition of the continuous component.
So, the main question becomes:
Given a measure $\gamma$ such that $\widehat{\gamma}$ is continuous, how can we identify $\gamma_1, \gamma_2$ such that $\widehat{\gamma_1}$ is a.c. and $\widehat{\gamma_2}$ is s.c.? Note that $\gamma_1, \gamma_2$ might actually not be measures. This leads to:
The Question
If we fix a compact set $K \subset \mathbb R^d$, can we get a nice description of the set
$$ \{ \widehat{f} | f \in L^1(\mathbb R^d), \operatorname{supp}(f) \subset K \} \,?$$
Note that for the type of problem I am interested, convolving $f$ with a continuous compactly supported function is OK. So any extra restriction on $f$ which can be obtained by convolutions with a class of compactly supported continuous functions (for example $f$ is continuous) is OK. Also, the condition $ \operatorname{supp}(f) \subset K$ can be replaced by $f$ compactly supported.
Best Answer
The Fourier transform of a function with compact support is automatically holomorphic. This support can be deduced from growth conditions on the former, at least in the case when it is a euclidean ball and the function is square integrable. This is the celebrated Payley-Wiener theorem, upon which there is a comprehensive Wikipedia article for starters.