Approximation of functions of class \(C(\epsilon)\) by subsequences of de la Vallée-Poussin sums (Q1177816)

Let \(C\) be the space of continuous \(2\pi\)-periodic functions \(f\) with the uniform norm \(\| f\|\), \(S_ n(f)\) be the partial trigonometric sums of order \(n\) and \(\sigma_{np}(f)={1\over p+1}\sum^ n_{\nu=n- p}S_ \nu(f)\), \(0\leq p\leq n\), \(n=0,1,\ldots,\) be the de la Vallée- Poussin sums \((\sigma_{n0}(f)=S_ n(f)\), \(\sigma_{nn}(f)=\sigma_ n(f)- \hbox{Fej\'er sums})\). Denote by \(E_ n(f)\) the best approximation of a function \(f\) by trigonometric polynomials of order not exceeding \(n\). \(C(\varepsilon)=\{f\in C:E_ n(f)\leq\varepsilon_ n\), \(n=0,1,\ldots\}\), where \(\varepsilon=\{\varepsilon_ n\}^ \infty_{n=0}\) is a sequence of a real numbers, \(\varepsilon_ n\downarrow 0\), \(n\to\infty\); suppose that \(e(C(\varepsilon),\sigma_{np})=\sup\{\| f-\sigma_{np}(f)\|:f\in C(\varepsilon)\}\). \textit{S. B. Steckin} [Anal. Math. 4, 61-74 (1978; Zbl 0393.41009)] and \textit{W. Dahmen} [Mat. Zametki 23, 671-683 (1978; Zbl 0385.42001)] independently proved that \[ e(C(\varepsilon),\sigma_{np})\asymp\sum^ n_{\nu=0}{\varepsilon_{n-p+\nu}\over p+\nu+1}. \] In the present paper the author proves that for every sequence \(\varepsilon_{n}\), \(\varepsilon_{n} \downarrow 0\), there exists a function \(f_{*}\in C(\varepsilon)\) which doesn't depend on \(n\) and \(p\) and for which \[ \| f_{*}-\sigma_{np}(f_{*})\| \gg \sum_{\nu=0}^{n} {\varepsilon_{n-p+\nu} \over p+\nu +1} \;(0\leq p\leq n,\;n=1,2,\ldots) \] is true. In this statement the fact that \(f_ *\) doesn't depend on \(n\) and \(p\) is essential.