On Certain Families of Generalized N¨orlund Methods and Power Series Methods

We consider families A 4of summability methods which have similar features a in their construction as the family of Ces`aro methods C, a.. 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 wR. Kiesel, Math. Z. 214 1993., 273]286; R. Kiesel and U. Stadtm¨uller, Canad. J. Math. 46, 1994., 982]994; R. Kiesel, Math. Nachr. 176 1995., 129]138x for certain families of generalized N¨orlund methods and Abel-type power series methods are combined with the results of the papers wA. Tali, Tartu Riikl. U¨l. Toimetised 960 1993., 117]138; M. M¨uristaja and A. Tali, Acta et Comment. Uni¨. Tartuensis Math. 1 1996., 93]103x in order to extend them, to improve the arguments used in wR. Kiesel, Math. Z. 214 1993., 273]286; R. Kiesel and U. Stadtm¨uller, Canad. J. Math. 46 1994., 982]994x and to get more general convex families of summability methods. The main theorems deal with two-parameter families A 4of generalized N¨orlund methods Theorems 3.1]3.3..