Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts

Let $ L_p$ be the classical Lebesgue spaces of $ 2\pi $-periodic functions and $ E(f,X)_2$ the best approximation of  $ f$ by the space  $ X$ in  $ L_2$. For $ n\in \mathbb{N}$, $ B\in L_2$, the symbol $ \mathbb{S}_{B,n}$ stands for the space of functions  $ s$ of the form $\displaystyle s(x)=\sum _{j=0}^{2n-1}\beta _jB\Big (x-\frac {j\pi }{n}\Big ).$     In this paper, all spaces  $ \mathbb{S}_{B,n}$ are described that provide a sharp constant in several inequalities for approximation of classes of convolutions with a kernel  $ G\in L_1$. In particular, necessary and sufficient conditions are obtained under which the inequality $\displaystyle E\bigl (f,\mathbb{S}_{B,n}\bigr )_2\leq \vert c^\ast _{2n+1}(G)\vert\Vert\varphi \Vert _2$     is fulfilled. This inequality is sharp on the class of functions  $ f$ representable in the form $ f=G\ast \varphi $, $ \varphi \in L_2$. The constant $ \vert c^\ast _{2n+1}(G)\vert$ is the $ (2n+1)$th term of the sequence $ \{\vert c_l(G)\vert\}_{l\in \mathbb{Z}}$ of absolute values of the Fourier coefficients of  $ G$ arranged in nonincreasing order. In addition, easily verifiable conditions are indicated that suffice for the above inequality. Examples of kernels and extremal subspaces satisfying these conditions are provided.