Interpolation in a Bernstein space by means of approximation

We denote by $B_\sigma$ the Bernstein space of entire functions of exponential type $\leq\sigma$ bounded on the real axis. Let $\Lambda=\{z_n\}_{n\in\mathbb Z}$, $z_n=x_n+iy_n$, be a sequence such that $x_{n+1}-x_n\geq l>0$ and $|y_n|\leq L$, $n\in\mathbb Z$. We prove that for any sequence $A=\{a_n\}_{n\in~\mathbb Z}$ of bounded $a_n$, $|a_n|\leq M$, $n\in\mathbb Z$, there exists a function $f\in B_\sigma$ with $\sigma\leq\sigma_0(l,L)$ such that $f|_\Lambda=A$. We use a method of approximation by mean of functions from a Bernstein space.