Geodesic
Inverse approximation theorem on an infinite union of segments
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 30, Tome 290 (2002), pp. 168-176

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Let $E=\bigcup\limits^{\infty}_{n=-\infty}[a_n, b_n]$, where $a_n$ and $b_n$ satisfy $0$, $0$ $n=0,\pm1,\pm2$. 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 $f$. 
@article{ZNSL_2002_290_a7,
     author = {N. A. Shirokov},
     title = {Inverse approximation theorem on an infinite union of segments},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {168--176},
     publisher = {mathdoc},
     volume = {290},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_290_a7/}
}
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N. A. Shirokov. Inverse approximation theorem on an infinite union of segments. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 30, Tome 290 (2002), pp. 168-176. http://geodesic.mathdoc.fr/item/ZNSL_2002_290_a7/