Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density (fist done by Beurling \begin{equation*} \delta(\Lambda):=u.u.d.(\Lambda)=\lim_{r\rightarrow \infty}\frac{\bar{n}(r)}{r} \end{equation*} Without loss of generality assume that $0\in\Lambda$.

Now define \begin{equation*} f(z)=\lim_{R\rightarrow\infty}\bigg\{\prod_{0<|\lambda|<R}(1-\frac{z}{\lambda})\bigg\} \end{equation*} Furthermore, fix $a>\delta(\Lambda)$. I am interested in calculating a sharp bound of the form below (with $C$ and $m$ as sharp as possible)

\begin{equation*} |f(x+iy)|\le C(|x+iy|+1)^m e^{a|y|} \end{equation*} Without the sharp bounds this has been studied by Arne Beurling and has implications for interpolation of deltas type functions (note that f is 1 at 0 and 0 on other entries in $\Lambda$) using entire functions.

what restrictions? You realize that the condition involving only the limit allows you to chose absolutely any $\lambda_j$ and make the function as large as you wish in any finite domain, don't you? $\endgroup$ – fedja Oct 24 '13 at 23:17