Classes of convolutions with a singular family of kernels: Sharp constants for approximation by spaces of shifts

Let $ \sigma >0$, and let $ G,B\in L(\mathbb{R})$. The paper is devoted to approximation of classes of functions  $ f$ for every $ \varepsilon >0$ representable as $\displaystyle f(x)=F_{\varepsilon }(x)+ \frac {1}{2\pi }\int _{\mathbb{R}}\varphi (t)G_{\varepsilon }(x-t) dt,$     where $ F_{\varepsilon }$ is an entire function of type not exceeding  $ \varepsilon $, $ G_{\varepsilon }\in L(\mathbb{R})$, and $ \varphi \in L_p(\mathbb{R})$. The approximating space  $ \mathbf S_B$ consists of functions of the form $\displaystyle s(x)=\sum _{j\in \mathbb{Z}}\beta _jB\Big (x-\frac {j\pi }{\sigma }\Big ).$     Under some conditions on $ G=\{G_{\varepsilon }\}$ and  $ B$, linear operators $ {\mathcal X}_{\sigma ,G,B}$ with values in  $ \mathbf S_B$ are constructed for which $ \Vert f-{\mathcal X}_{\sigma ,G,B}(f)\Vert _p\leq {\mathcal K}_{\sigma ,G}\Vert\varphi \Vert _p$. For $ p=1,\infty $ the constant $ {\mathcal K}_{\sigma ,G}$ (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space  $ \mathbf S_B$. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.