Steinhaus Type Theorem for Nörlund-(M, λn) and Nörlund-Euler-(M, λn) Methods of Summability in Non-Archimedean Fields

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E. Muthu Meena Lakshmanan, K. Suja


In the present research paper, an investigation is undertaken of Steinhaus type theorems for Nörlund-(M, λn) and Nörlund-Euler(M, λn) method of summability in K, a complete non-trivially valued Non-Archimedean field. The conditions for statistical summability for those matrices are discussed in such fields K. The consistency of Nörlund-(M, λn) method of summability is investigated when different sequences are used in the summation process. Further, the relation between Nörlund-Euler(M, λn) summable and its statistical summability is also established.

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  1. G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, New York, 1964.
  2. E.A. Aljimi, V. Loku, Generalized Weighted Norlund-Euler Statistical Convergence, Int. J. Math. Anal. 8 (2014), 345–354.
  3. E.A. Aljimi, E. Hoxha, V. Loku, Some Results of Weighted Norlund-Euler Statistical Convergence, Int. Math. Forum. 8 (2013), 1797–1812.
  4. A.F. Monna, Sur le Theoreme de Banach-Steinhaus, Indag. Math. (Proc.) 66 (1963), 121–131.
  5. V. Loku and E. Aljimi, Weighted Norlund-Euler λ-Statistical Convergence and Application, J. Math. Anal. 9 (2018), 95–105.
  6. P.N. Natarajan, Characterization of a Class of Infinite Matrices With Applications, Bull. Austral. Math. Soc. 34 (1986), 161–175.
  7. P.N. Natarajan, The Steinhaus Theorem for Toeplitz Matrices in Non-Archimedean Fields, Ann. Soc. Math. Pol. Ser. I: Comment. Math. 20 (1978), 417–422.
  8. P.N. Natarajan, A Steinhaus Type Theorem, Proc. Amer. Math. Soc. 99 (1987), 559–562.
  9. P.N. Natarajan, On No¨rlund Method of Summability in Non-Archimedean Fields, Analysis. 2 (1994), 97–102.
  10. P.N. Natarajan, Weighted Means in Non-Archimedean fields, Ann. Math. Blaise Pascal. 2 (1995), 191–200.
  11. P.N. Natarajan, Some Steinhaus Type Theorems Over Valued Fields, Ann. Math. Blaise Pascal. 3 (1996), 183–188.
  12. P.N. Natarajan, Some Properties of the (M,λn) Method of Summability in Ultrametric Fields, Int. J. Phys. Math. Sci. 2 (2012), 169–176.
  13. P.N. Natarajan, Steinhaus Type Theorems for Summability Matrices in Ultrametric Fields, Int. J. Phys. Math. Sci. 3 (2013), 64–69.
  14. P.N. Natarajan, Cauchy Multiplication of (M, λn) Summable Series in Ultrametric Fields, International J. Phy. Math. Sci. 3 (2013), 51–55.
  15. R.E. Powell, S.M. Shah, Summability Theory and Its Applications, Van Nostrand-Reinhold Company, London, 1972.
  16. V.K. Srinivasan, On Certain Summation Processes in the p-Adic Field, Indag. Math. (Proc). 68 (1965), 319–325.
  17. K. Suja, V. Srinivasan, On Statistically Convergent and Statistically Cauchy Sequences in Non-Archimedean Fields, J. Adv. Math. 6 (2014), 1038–1043.