The approximation of the analytic functions of the repeated sums of Vallée Poussin (Q2880020)

In the present paper some inequalities for the characteristics of approximation of functions are obtained. Suppose that \(\psi(k)\) is an arbitrary function of natural variable, and \(\beta\) is a real number. If some series depending on \(\psi\) and \(\beta\) is a Fourier series of some function \(f_{\beta}^\psi,\) it is called the \((\psi, \beta)\)--de\-ri\-va\-tive of \(f.\) Denote the set of all continuous functions having \((\psi, \beta)\)--de\-ri\-va\-tives by \(C_{\beta}^{\psi}.\) If \(f\in C_{\beta}^{\psi}\) and, in addition, \(\int\limits_{-\pi}^\pi f_{\beta}^\psi dx=0\) and \(\text{ ess\,}\sup\,| f_{\beta}^\psi(x)| \leq 1,\) we write \(f\in C_{\beta,\infty}^{\psi}.\) In what follows, we consider the case \(\psi(k)=e^{-\alpha k},\) \(k=1,2,\ldots,\) and \(\alpha\in {\mathbb R}.\) Denote by \(C_{\beta,\infty}^{\alpha}:= C_{\beta,\infty}^{\psi}\) with \(\psi\) as above. The repeated arithmetic mean of Fourier sums are introduced in the paper and denoted by \(V_{n, p}^2(f, x).\) Let \(\alpha>0,\) and \(\beta\in {\mathbb R}.\) Then, in the notions given above, the asymptotic formula for the quantity \({\mathcal E} \left(C_{\beta, \infty}^{\alpha}, V_{n, p}^2(f, x)\right):= \sup\limits_{f\in C_{\beta, \infty}^{\alpha}}\| f(x)-V_{n, p}^2(f, x)\|_{C}\) is obtained. The results of the paper can be applied to many problems of mathematical analysis.