Let $F,H:\mathbb{C}\to\mathbb{C}$ be entire functions of mean exponential type and of completely regular growth. Assume further that the indicator diagrams $I_F$ and $I_H$ are on the imaginary axis and separated, e.g., $I_F=\imath[a,b]$ and $I_H=\imath[b+1,c]$.

**Question:** Does $F+H \in L^2(\mathbb{R})$ imply $F\in L^2(\mathbb{R}
)$?

In other words, can $F$ and $H$ interfere on $\mathbb{R}$ in a way that $F+H\in L^2(\mathbb{R}^2)$ without $F$ or $H$ being $L^2(\mathbb{R})$?

**Discussion:** Motivation behind this question comes from real analysis. If we assume in addition that $F,H$ are polynomially bounded on $\mathbb{R}$ then they are Fourier transforms of compactly supported distributions $u_F,u_H$ on $\mathbb{R}$, and $\mathrm{supp}\,u_F=\imath I_F$, $\mathrm{supp}\,u_H=\imath I_H$ (Paley-Wiener, Plancherel-Polya). Qualitatively, the behaviour at infinity of $F$ and $H$ reflect the local regularity of $u_F$ and $u_H$. If $F+
H\in L^2(\mathbb{R})$ then $u_F+u_L\in L^2(\mathbb{R})$, i.e., the sum of the two distributions is locally integrable. However, since $\mathrm{supp}\,u_F\cap\mathrm{supp}\,u_H=\emptyset$, adding $u_H$ to $u_F$ cannot change the local regularity of $u_F$, and we expect $u_F$ to be locally integrable as well, which entails $F\in L^2(\mathbb{R})$. In the above question we do not assume that $F$ and $H$ are polynomially bounded on $\mathbb{R}$, but we let $I_F$ and $I_H$ be at a positive distance. In this case $F$ and $H$ can be thought of as Fourier transforms of analytical functionals, but the latter objects do not have the notion of compact support, and the analogy doesn't work.

Thank you.

**Edit:** For an entire function $F$ of mean exponential type, the indicator function can be defined as
$$
h_F(\theta)=\limsup_{r\to\infty}\frac{\log|f(re^{\imath\theta})|}{r}
$$
and indicates the rate of growth in direction $\theta$. It can be shown that there exists a compact convex set $I_T\subset\mathbb{C}$ (called the indicator diagram of $F$) such that $h_F$ is exactly the support function of $I_F$,
$$
h_F(\theta)=\sup_{z\in I_F}\mathrm{Re}\left[ze^{-\imath\theta}\right].
$$
Intuitively, the thickness of the projection of the set $I_F$ on the ray $\theta$ shows the rate of growth of $F$ along that ray.