Inverse approximation theorem on an infinite union of segments

Let $E=\bigcup\limits^{\infty}_{n=-\infty}[a_n, b_n]$, where $a_n$ and $b_n$ satisfy $0<c_1\le b_n-a_n\le c_2$, $0<c_3\le a_{n+1}-b_n\le c_4$ $n=0,\pm1,\pm2$. Denote by $B_{\sigma}$ the class of all entire functions of exponential type $\le\sigma$ bounded on the real axis. Under certain assumptions on the rate of approximation on $E$ of a bounded function $f$ by functions in $B_{\sigma}$ ($\sigma$ varies), we get some information about the smoothness of $f$.