• On the relation of generalized Valiron summability to Cesàro summability

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/090/03/0181-0193

    • Keywords

       

      Generalized Valiron summability; Boral summability; Rajagopal’s theorem

    • Abstract

       

      A family (Vak) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (Bα,γ), (Eρ, (Tα), (Sβ) and (Va) are members of this family. In §3 some properties of the (Bα,γ) and (Vak) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (Vak) 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.If sn (n ≥ 0) isa real sequence satisfying$$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$, and if sns (Vak) thensn → s (C, 2ρ).

    • Author Affiliations

       

      V K Krishnan1

      1. St. Thomas College, Trichur - 680 001, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

© 2017-2019 Indian Academy of Sciences, Bengaluru.