On Certain Families of Generalized Nörlund Methods and Power Series Methods

ISSN： 0022-247X

Source： Journal of Mathematical Analysis and Applications, Vol.238, Iss.1, 1999-10, pp. : 44-66

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Abstract

We consider families {A_α} 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. Toimetised 960 (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 {A_αβ} of generalized Nörlund methods (Theorems 3.1–3.3).