Nikol'skii theorem for kernels satisfying the more general condition than \(A_n^*\) (Q2901753)

The present paper studies the possibility of the approximation of functions by trigonometric polynomials. The well-known theorem of Nikol'skii states some equalities between the best approximation of the function, realized in the view of convolution of functions \(\varphi\) and \(K,\) and the best approximation of the kernel \(K(t),\) provided that some condition \(A_n^*\) holds for the \(K\). Now the author considers a weaker condition \(B_n^*\) consisting in the following. A function \(K\in L_1\) satisfies the condition \(B_n^*\) if and only if there exists a trigonometric polynomial \(T^*\) of the degree \(\leq n-1,\) a function \(\varphi_*\in L_{\infty}\) and some \(n_*\geq n\) such that almost everywhere \(| \varphi_*(t)| \leq 1,\) \(\varphi_*(t)(K(t)-T^{*}(t))=| K(t)-T^*(t)| \) and \(\varphi_*(t+\pi/n_*)=-\varphi_*(t)\). The statement of Nikol'skii's theorem is proved by the author for such kernels \(K\). Examples of kernels satisfying the condition \(B_n^*\) are also constructed in the work.