Entire function that grows along one axis and decays along the other I would like to find an entire function on the complex plane that grows with $|\mathrm{Re}(z)|$ no faster than $e^{\alpha |\mathrm{Re}(z)|}$ (for some fixed positive $\alpha$), and decays as fast as possible in both directions along the $\mathrm{Im}(z)$ axis.
As an example, we could consider $\cosh(\alpha z)$. This grows like $e^{\alpha |\mathrm{Re}(z)|}$ along the real axis and oscillates along the imaginary axis. By averaging over different $\alpha$'s, we can make it decay like $1/\mathrm{Im}(z)$ in the imaginary direction,
$\int_0^\alpha d\alpha' \cosh(\alpha' z)  = \frac{\sinh(\alpha z)}{z}.$
And by averaging again over $\alpha$, we can make it decay like $1/\mathrm{Im}(z)^2$,
$\int_0^\alpha d\alpha' \frac{\sinh(\alpha' z)}{z} = \frac{\cosh(\alpha z) - 1}{z^2}.$
However, beyond this point the averaging trick doesn't make it decay any faster in the imaginary direction.
Is there an entire function that grows at most like $|\mathrm{Re}(z)|$ in the real directions and decays faster than $1/z^2$ in the imaginary directions?
 A: The answer to this question is contained in the famous theorem of Beurling and
Malliavin. It is usually stated with decrease on the real axis (rather than imaginary axis as in your question, but you make the change $z\mapsto iz$ to obtain what you need). First of all, the growth restriction is
$$\log|f(z)|\leq (\alpha+o(1))|z|$$
this is weaker than your growth restriction. This is called "exponential type $\alpha$". Then suppose that $|f|$ is bounded
on the real axis (or belongs to $L^2$, this can be relaxed). Then it is not
difficult to see that
$$\int_R\frac{|\log|f(x)||}{1+x^2}dx<\infty.$$
This gives the maximal rate of decrease of $|f|$ on the real axis.
Under some additional conditions, this rate is best possible.
Beurling and Malliavin give two such conditions. Let $w(x)\geq 1$ be a weight function satisfying
$$\int_R\frac{\log w(x)}{1+x^2}dx<\infty.$$
This is called the "logarithmic integral". 
Then there exists an entire function $f$ of (arbitrarily small) exponential type $\alpha$ such that
$fw$ is bounded on the real line, if $w$ satisfies any of the following additional regularity conditions:
a) $\log w$ is uniformly continuous, or
b) $w$ is itself a function of (some) exponential type.
This theorem is difficult. But there is a much simpler result due to Wiener and Paley: it is sufficient that $w$ is even and increasing on the positive ray,
and the logarithmic integral is convergent.
And of course it is not a problem to construct such function decreasing as
$|x|^{-n}$ for every $n$: just divide out as many zeros of $\cos$ as you need.
References: MR0147848 Beurling, A.; Malliavin, P. On Fourier transforms of measures with compact support. Acta Math. 107 1962 291–309.
MR1430571
Koosis, Paul
Leçons sur le théorème de Beurling et Malliavin. (French) [Lessons on the Beurling-Malliavin theorem] Université de Montréal, Les Publications CRM, Montreal, QC, 1996
This book is a very nice discussion of the whole subject, with alternative proofs.
