On a generalization of some theorems on the smoothness of the sum of trigonometric series

Abstract

In this paper we have considered the trigonometric series $$\sum_{m\in \mathbb{Z}}c_{m}e^{imx},$$ where $(c_{m})_{\in \mathbb{Z}}$ is a sequence of complex numbers such that $$\sum_{m\in \mathbb{Z}}|m|^{r-1}|c_{m}|<+\infty,\,\,\,\, (r=1,2,\dots ).$$ Then the $(r-1)$-th derivative of the trigonometric series converges absolutely and uniformly. If we denote by $f(x)$ the sum function of such trigonometric series, then its $(r-1)$-th derivative $f^{(r-1)}(x)$ obviously is a continuous one. We have given sufficient conditions in terms of some means of $(c_{m})_{\in \mathbb{Z}}$ to ensure that $f(x)$ belongs to one of the classes $W^{r}(\alpha )$ or $w^{r}(\alpha )$ for $0<\alpha \leq 2$. The results of Krizs\'{a}n and M\'oricz obtained in \cite{KM} and those of Zygmund obtained in \cite{Z} are particular results of ours.