Degree of approximation of functions in the Hölder metric by Borel's means (Q5904093)

Let \(s_n(f; x)\) be the partial sum of the first \((n+1)\) terms of the Fourier series of \(f\in L_{2\pi}\). We write \[ \sigma_n(x)= {1\over n} \sum^n_{k= 0} s_k(f;x)\quad (n>0).\tag{1} \] Recently, \textit{R. Mohanty} and \textit{B. K. Ray} [Bull. Calcutta Math. Soc. 86, No. 1, 89-98 (1994; Zbl 0856.42009)] have studied the convergence and absolute convergence of the series \[ \sum^\infty_{n=1} {\sigma_n(x)- f(x)\over n}.\tag{2} \] For real \(x\), write \[ \phi(t)= {1\over 2}\{f(x+ t)+ f(x- t)- 2f(x)\},\quad c_0= -{2\over\pi} \int^\pi_0{(\pi u- u^2)\over (2\sin{1\over 2}u)^2} \phi(u)du, \] \[ T_n(x)= \sum^n_{k=1} {\sigma_k(x)- f(x)\over k}\quad (n\geq 1)\quad\text{and}\quad T_0(x)= 0. \] In this paper, the authors first show the Fourier character of the series (2) and then obtain the following estimate in Hölder metric: Let \(0\leq\beta< \alpha\leq 1\) and let \(f\in H_\alpha\). Then \[ \| B_p(T)+\textstyle{{1\over 2}} c_0\|_\beta= O(1) \begin{cases} p^{\beta- \alpha}\quad &(\alpha-\beta\neq 1)\\ p^{-1}\log p\quad & (\beta= 0, \alpha= 1),\end{cases} \] where \(B_p(T;x)\) is the Borel-exponential mean of the sequence \(\{T_n(x)\}^\infty_{n=0}\). Remark. It may be observed that the above estimate in the case \(\alpha-\beta\neq 1\) is sharper than the one obtained by these authors [J. Math. Anal. Appl. 219, No. 2, 279-293 (1998; Zbl 0914.42002)].