is the domain introduced in Section \[Lim\]. An entire function $f$ in $\operatorname*{\mathbb{C}}$ is said to be of *finite order* if there exist $0 \leq \rho < \infty$ and $C_1,C_2 > 0$ such that $|f(z)| \leq C_1 e^{C_2 |z|^{\rho}}$ for all $z \in \operatorname*{\mathbb{C}}$. The infimum of such $\rho$ that the above $C_1,C_2$ exist is called the *order* of $f$. If $f$ has an order $\rho$, then $f$ is said to be of *finite type* if $\overline{\lim}_{a \to \infty}a^{-\rho}\log(\sup_{|z|\leq a}|f(z)|) < \infty$. The value of the limit superior is called the *type* of $f$. We refer the reader to [@Lev] for more information on entire functions.

Let $\mu$ be a probability measure on $\operatorname*{\mathbb{R}}$. The following conditions are equivalent.

-   $\mu$ belongs to $\mathcal{P}_1$.

-   $G_\mu(\frac{1}{z})$ defined in $\operatorname*{\mathbb{C}_{+}}$ has analytic continuation to $\{z \in \operatorname*{\mathbb{C}}; |z| < r \}$ for some $r > 0$.

-   $F_\mu(\frac{1}{z}) - \frac{1}{z}$ defined in $\operatorname*{\mathbb{C}_{+}}$ has analytic continuation to $\{z \in \operatorname*{\mathbb{C}}; |z| < r \}$ for some $r > 0$.

-   $F_\mu^{-1}(\frac{1}{z})-\frac{1}{z}$ defined in $\Omega^{-1}$ has analytic continuation to $\{z \in \operatorname*{\mathbb{C}}; |z| < r \}$ for some $r > 0$.

-   The Fourier transform $\mathcal{F}_\mu(z)$ defined in $z > 0$ is 