Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line

For the classes $L^r_2(\mathbb{R})$, $r\in \mathbb{Z}_{+}$, we establish the upper and lower bounds for the quantities
$$ \chi_{\sigma,k,r,\mu,p}(\psi,t):=\sup\biggl\{\mathcal{A}_{\sigma} (f^{(r-\mu)})\Bigm/\biggl(\int_0^t \omega^p_k(f^{(r)},\tau) \psi(\tau)\,d\tau\biggr)^{1/p}:f \in L^r_2(\mathbb{R})\biggr\}, $$
where $\mu, r \in \mathbb{Z}_{+}$, $\mu \le r$, $k \in \mathbb{N}$, $0< p \le 2$, $0< \sigma <\infty$, $0<t \le \pi/\sigma$, and $\psi$ is a nonnegative, measurable function summable on the closed interval $[0,t]$ and not equivalent to zero. In the cases $\chi_{\sigma,k,r,\mu,p}(1,t)$, where $\mu\in \mathbb{N}$, $1/\mu\le p \le 2$, and $\chi_{\sigma,k,r,\mu,2/k}(1,t)$, where $0<t \le \pi/(2 \sigma)$, we obtain the exact values of these quantities. We also obtain the exact values of the average $\nu$-widths of classes of functions defined in terms of the modulus of continuity $\omega^{*}$ and the majorant $\Psi$.