Best mean-square approximation of functions defined on the real axis by entire functions of exponential type

Abstract

Exact constants in Jackson-type inequalities are calculated in the space $L_2 (\mathbb{R})$ in the case where the quantity of the best approximation $\mathcal{A}_{\sigma}(f)$ is estimated from above by the averaged smoothness characteristic $\Phi_2(f, t) = \cfrac 1t \int^t_0||\Delta^2_h(f)||dh$. We also calculate the exact values of the average $\nu$-widths of classes of functions defined by $\Phi_2$.