Some Approximating Properties of Cesàro Means of the Series $\sigma \left[ f \right]$ and $\bar \sigma \left[ f \right]$

Abstract

By now, a great deal of fundamental and profound results has been established concerning approximating properties of means of Cesaro, Poisson-Abel, Hardy, and of general linear means of the series $\sigma \left[ f \right]$ and $\bar \sigma \left[ f \right]$. Most of them are set forth in monographs and proceedings of international conferences, schools and symposia. Even several monographs would not suffice for a complete account of the results obtained in this direction. Therefore below we will survey only those of them which are necessary in the study of the questions treated in this chapter. We note, however, that the monographs by Alexits [2], Akhiezer [1], Bari [2], Vallée-Poussin [1], Jackson [3], Dzyadyk [1], Zygmund [15], Korneïchuk [1], Leindler [1], Lorentz [2], Natanson [1], Nikol’skiï [5], Stepanets [1], Timan [1], as well as the papers by Korneïchuk [2] and Tikhomirov [1], give the reader an idea of the most important investigations inthis field.