by Fourier transforms results in a kind of Paley-Wiener theorem.

Complex moments {#Genmom}
===============

We introduce a class $\mathcal{P}_1$ of probability measures $\mu$ of the forms $$\label{decomp}
\mu= \nu + \lambda,$$ where $\nu$ and $\lambda$ satisfy the following properties.

-   $\nu$ is a positive measure with compact support.

-   There exist $0 < r < R$ and a real analytic function $p(x) = \sum_{n=2}^\infty \frac{a_n}{x^n}$ which is absolutely convergent for $|x|>r$ and is non-negative on $|x|\geq R$ such that $\lambda(dx) = p(x)\textbf{1}_{|x|\geq R}(x)dx$.

We also define $\mathcal{P}_2$ consisting of $\mu = \nu + \lambda$, where $\nu$ is a positive measure with finite second moment and $\lambda$ satisfies the property (A2).

The following measures satisfy the property (A2).

-   Cauchy distributions $\lambda = \mu_{a,b}$ defined on $|x| \geq R$ with $R > |a| + b$.

-   $\lambda = \frac{1}{\sqrt{1 + x^4}}dx$ on $|x| \geq R$ with $R > 1$.

-   $\lambda =  \frac{P(x)}{Q(x)}dx$ on $(-\infty,-R] \cup [R, \infty)$ for sufficiently large $R>0$, where $P(x)$ and $Q(x)$ are polynomials satisfying that $\text{deg~}P(x)+2 \leq \text{deg~}Q(x)$ and $P(x)$, $Q(x) \geq 0$ for $|x| > R$.

-   $\lambda = \frac{1}{(1+x^{2n})^{\frac{p}{n}}}dx$ on $|x| \geq R$ with $R>1$, where $n \geq 1$ and $p \geq 1$ are 