Abstract:
We show that for negative $\alpha$ Sunouchi's formula
\begin{gather*}
H_n(f,\alpha,\beta,x)=\frac1{A^\beta_n}\sum_{k=0}^nA_{n-k}^{\beta-1}|f(x)-\sigma_k^\alpha(f,x)|,\\
\alpha>-\frac12,\quad\beta>\frac12,
\end{gather*}
becomes false, where $\sigma_k^\alpha(f,x)$ is the $(C,\alpha)$ mean of the Fourier series
for the function $f(x)\in\mathrm{Lip}\,\gamma$, $0<\gamma<1$. A bound is given for
$H_n(f,\alpha,\beta,x)$ for all $\alpha>-1$, $\beta>-1$, which for $\alpha+\beta>0$, $\alpha\geqslant0$, $\beta\geqslant0$,
coincides with the Sunouchi bound. The proof is by a method different from that of Sunouchi.
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