On the relation of generalized Valiron summability to Ces'aro summability

Abstract

A family (Va ~) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (Ba, 7), (Er (Ta), (SI~) and (Va) are members of this fami!y. In w some properties of the (Ba,~,) and (V~) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V~) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem. /f s n (n ~ 0) is a real sequence satisfying lira lim inf min (s, ~ sin" ~ e->0+ ~-~o ,.~<.~,,+ex/., \--m-P-./> o (p ~> 0), and if s n --* s (V~) then s, ---> s (C, 2p).