On Certain Families of Generalized Nörlund Methods and Power Series Methods

Abstract

We consider families {Aα} of summability methods which have similar features in their construction as the family of Cesàro methods (C,α). Abel-type power series methods will be added to those families and inclusion and Tauberian theorems will be proved. The Tauberian and inclusion theorems proved in the papers [R. Kiesel, Math. Z.214 (1993), 273–286; R. Kiesel and U. Stadtmüller, Canad. J. Math.46, (1994), 982–994; R. Kiesel, Math. Nachr.176 (1995), 129–138] for certain families of generalized Nörlund methods and Abel-type power series methods are combined with the results of the papers [A. Tali, Tartu Riikl. Ül. Toimetised960 (1993), 117–138; M. Müristaja and A. Tali, Acta et Comment. Univ. Tartuensis Math.1 (1996), 93–103] in order to extend them, to improve the arguments used in [R. Kiesel, Math. Z.214 (1993), 273–286; R. Kiesel and U. Stadtmüller, Canad. J. Math.46 (1994), 982–994] and to get more general convex families of summability methods. The main theorems deal with two-parameter families {Aαβ} of generalized Nörlund methods (Theorems 3.1–3.3).