A note on a theorem of Sunouchi

Abstract

We show that for negativeα Sunouchi's formula

$$\begin{gathered} H_n (f,\alpha ,\beta ,x) = \frac{1}{{A_n^\beta }}\sum\nolimits_{k = 0}^n {A_{n - k}^{\beta - 1} } |f(x) - \sigma _k^\alpha (f,x)|, \hfill \\ \alpha > - \frac{1}{2},\beta > \frac{1}{2}, \hfill \\ \end{gathered}$$

becomes false, where σ ^α_k (f, x) is the (C,α) mean of the Fourier series for the functionf(x) ε Lipγ, 0<γ<1. A bound is given for H_n(f, α,β, x) for allα > -1,β> -1, which forα + β > 0, α≥ 0,β ≥0, coincides with the Sunouchi bound. The proof is by a method different from that of Sunouchi.