Special Summability Methods

  • P. N. NatarajanEmail author


In the current chapter, we introduce some special methods of summability, viz. the Abel method, the Weighted Mean method, the Euler method and the \((M, \lambda _n)\) or Natarajan method, and study their properties extensively. The connection between the Abel method and the Natarajan method is brought out.


The Weighted Mean method Hardy Móricz and Rhoades The \((M, \lambda _n)\) or Natarajan method Y-method Consistent Translative Inclusion theorem Equivalence theorem The Abel method Product theorem The Euler method Invertible 


  1. 1.
    Peyerimhoff, A.: Lectures on Summability. Lecture Notes in Mathematics, vol. 107. Springer, Berlin (1969)zbMATHGoogle Scholar
  2. 2.
    Natarajan, P.N.: A generalization of a theorem of Móricz and Rhoades on Weighted means. Comment. Math. Prace Mat. 52, 29–37 (2012)zbMATHGoogle Scholar
  3. 3.
    Hardy, G.H.: A theorem concerning summable series. Proc. Cambridge Philos. Soc. 20, 304–307 (1920-21)Google Scholar
  4. 4.
    Móricz, F., Rhoades, B.E.: An equivalent reformulation of summability by weighted mean methods. revisited. Linear Algebra Appl. 349, 187–192 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Natarajan, P.N.: Another theorem on weighted means. Comment. Math. Prace Mat. 50, 175–181 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Maddox, I.J.: Elements of Functional Analysis. Cambridge University Press, Cambridge (1977)zbMATHGoogle Scholar
  7. 7.
    Natarajan, P.N.: On the \((M, \lambda _n)\) method of summability. Analysis (München) 33, 51–56 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Natarajan, P.N.: A product theorem for the Euler and the Natarajan methods of summability. Analysis (München) 33, 189–195 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Natarajan, P.N.: New properties of the Natarajan method of summability. Comment. Math. Prace Mat. 55, 9–15 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hardy, G.H.: Divergent Series. Oxford (1949)Google Scholar
  11. 11.
    Powell, R.E., Shah, S.M.: Summability Theory and Applications. Prentice-Hall of India (1988)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Formerly of the Department of MathematicsRamakrishna Mission Vivekananda CollegeChennaiIndia

Personalised recommendations