Convergence of the Cesàro means of trigonometric Fourier series
Let f be a 2\(\pi\)-periodic function. Its maximal modulus of continuity \({\hat \omega}{}_ 1(\delta,f)\) is defined by \({\hat \omega}_ 1(\delta,f)=\sup_{\Delta}\{\sup_{| h| \leq \delta}\int_{\Delta}| f(x+h| \Delta |)-f(x)| dx\}, 0<\delta \leq 1\), the supremum being taken over all intervals \([a,b]=\Delta \subset {\mathbb{R}}\), \(| \Delta | =b-a<\pi\). The author proves that if f is 2\(\pi\)-periodic and continuous and \(\int^{1}_{0}\frac{{\hat \omega}_ 1(t,f)}{t^{1+\alpha}}dt<\infty\), \(0\leq \alpha<1\), then the Fourier series for f is uniformly (C,-\(\alpha)\) summable. The hypothesis on the growth of \({\hat \omega}{}_ 1\) cannot be weakened. This result improves a result of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Math. Z. 28, 612-634 (1928; JFM 54.0301.03)] and extends earlier results of \textit{E. A. Sevastyanov} [Mat. Sb., Nov. Ser. 114, 583-610 (1981; Zbl 0476.42002); Anal. Math. 1, 283-295 (1975; Zbl 0314.42008] from \(\alpha =0\) to \(\alpha<1\).
scientific article; zbMATH DE number 3851695