On the best mean square approximations by entire functions of exponential type in $L_2(\mathbb R)$ and mean $\nu$-widths of some functional classes

We consider some extremal problems of approximation theory of functions at the whole real axis $\mathbb R$ by entire functions of the exponential type. In particular, we find the exact values of the mean $\nu$-widths of classes of functions, defined by the moduli of continuity of $m$th order $\omega_m$ and majorants $\Psi$ satisfying the special type of restriction.