On exact constants for matrix summation methods
The author gives exact constants for the values \[ \begin{aligned} A(\Lambda, \alpha_n) &:=\sup_{\substack{ f\in C_{2\pi}\\ f\neq \text{const} }} {{|f(x)- U_n (f, \Lambda, x) |_{C_{2\pi}}} \over {\omega (f, \alpha_n)}} \\ \text{and} A_1 \biggl( \Lambda, {\pi \over n} \biggr) &:=\sup_{\substack{ f'\in C_{2\pi}\\ f'\neq \text{const} }} {{|f(x)- U_n (f, \Lambda, x) |_{C_{2\pi}}} \over {\omega (f', {\pi \over n})}}, \end{aligned} \] where \(\alpha_n:=n^{-1} \gamma \log n\), \(\gamma> 0\) and \[ U_n (f, \Lambda, x):={a_0 \over 2}+\sum^n_{\nu =1} \lambda_\nu^{(n)} (a_\nu \cos \nu x+b_\nu \sin \nu x). \] E.g. Theorem 1 states: If \(\{\lambda_\nu^{(n)} \}\) is convex down respecting \(\nu\) and \(\lambda_\nu^{(n)}\to 1\), \(n\to \infty\), then \[ \begin{multlined} A(\Lambda, \alpha_n)=1+{2\over {\pi \gamma}} (1- \lambda_1^{(n)}) \biggl(n- {{n\ln (\gamma \ln n)} \over {\ln n}} \biggr)+\\ O\biggl( {n\over {\ln n}} (1- \lambda_1^{(n)}) \biggr)+O\biggl( {n^2 \over {\ln^2 n}} \sum^n_{\nu =0} \Delta^2 \lambda_\nu^{(n)} {1\over {\nu+1}} \biggr) \qquad \text{as} \quad n\to \infty.\end{multlined} \] {}.
scientific article; zbMATH DE number 923549