Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups
Although summability for Vilenkin-Fourier series in the bounded case has been known for nearly a quarter century [see \textit{J. Pal} and \textit{P. Simon}, Acta Math. Acad. Sci. Hung. 29, 155-164 (1977; Zbl 0345.42011)], the unbounded case has defied resolution until now. In this interesting paper, the author proves that if \(f\in L^p(G)\) for \(p> 1\), where \(G\) is \textit{any} Vilenkin group (of bounded type or not), then the Vilenkin-Fourier series of \(f\) is \((C,1)\) summable to \(f\) almost everywhere. The hang-up had been that the Fejér kernels for the unbounded case where \textit{not} uniformly bounded in the \(L^1\) norm. The author gets around this dilemma by using an ingenious decomposition of the maximal Cesàro operator into four pieces which isolate the difficulties one at a time.
scientific article; zbMATH DE number 1389107
scientific article