Analytic functions vanishing (sub)exponentially at infinity Let $f$ be an analytic function in the upper complex half-plane and continuous up to the real axis, and let $a>0$. Suppose that the function
\begin{equation}
\zeta\in\mathbb{C}^+\rightarrow f(\zeta)\mathrm{e}^{-ia\zeta}\in\mathbb{C}
\end{equation}
is itself bounded. Intuitively, since the absolute value of the exponential grows as $|z|\to\infty$, this requires $f$ to decay at least exponentially, with exponent larger than $a$, at $|z|\to\infty$; for example, any function like $f(\zeta)=\mathrm{e}^{ib\zeta}$, $b>a$ will do the trick, as well as any combination of such functions.
I wonder whether the class of analytic, bounded functions in the half-plane satisfying this condition is in fact larger and/or can be characterized somehow.
 A: Holomorphic functions with controlled growth usually appear in the theory of integral transforms of generalized function. Consider, for example, the class of holomorphic functions bounded on the right half plane by an exponential function i.e. such that
$$
\mathscr{LH}_a\triangleq\big\{ f\text{ is holomorphic for }\Re\zeta>-a \text{ and } |f(\zeta)|\le Ce^{-L|\zeta|},\; \Re \zeta>0\big\}.\label{1}\tag{1}
$$
for some $L>0$ (assuming nothing on the regularity of the function $f$ for $\Im \zeta=0$).
It can be proved that ([2] p. 400 and p. 403) an analytic function $f$ belongs to $\mathscr{LH}_a$ if and only if it is the Laplace transform of a Laplace hyperfunction: and the class \eqref{1} up to a counterclockwise rotation of $\pi/2$ of the domain of definition of its members , strictly includes the class of holomorphic functions bounded in the upper half plane and continuous on the real axis, i.e. if $f$ is bounded on the upper half plane and continuous on the real axis, then $f(-i\zeta)\in\mathscr{LH}_0$.
Apart from this "modern" characterization on this class of function, Torsten Carleman used functions bounded on the upper and the lower half plane to define his generalized Fourier transform: his result are collected in the monograph [1].
References
[1] Thorsten Carleman, L’intégrale de Fourier et questions qui s’y rattachent (French), Publications Scientifiques de l'Institut Mittag-Leffler, 1, Uppsala. 119 p. (1944), MR0014165, Zbl 0060.25504.
[2] Eungu Lee and Dohan Kim, "Laplace hyperfunctions", Integral Transforms and Special Functions, 19:6, 399-407 (2008), MR2426730, Zbl 1186.46042.
