Widths of certain classes of convolutions (Q791798)

Let \(W^{r,\alpha}_ X\), r, \(\alpha \in R^ 1\), \(r>0\) be the class of functions f belonging to a space X of 2\(\pi\)-periodic functions, which are representable in the form of convolution \(f(x)=K_{r,\alpha}*\phi(x)\), where \(K_{r,\alpha}(x)=\sum^{\infty}_{\nu =1}\nu^{-r}\cos(\nu x- \frac{\alpha \pi}{2}), \| \phi \|_ X\leq 1\), \(\int^{2\pi}_{0}\phi(x)dx=0\). The quantity \[ d_ n(W^{r,\alpha}_ X\quad,X)=\inf_{M_ n}\sup_{f\in W^{r,\alpha}_ X} \inf_{g\in M_ n}\| f-g\|_ X, \] n\(=1,2,...\), where \(M_ n\subset X\) are n-dimensional subspaces is called Kolmogorov's diameter of the class \(W^{r,\alpha}_ X\) in the space X. For the spaces \(X=L_ 1\) and \(X=L_{\infty}\) in the case of natural r and \(\alpha =r\) it is known the following estimate \[ d_{2n- 1}(W^{r,\alpha}_ X\quad;X)\geq \| K_{r,\alpha}*\phi_ n\|_{L_{\infty}}, \] where \(\phi_ n(x)=sign \sin nx.\) In the paper it is proved, that under the condition \(r=1\) this estimate is true for any \(\alpha \in R^ 1\).