TAUBERIAN THEOREM FOR GENERAL MATRIX SUMMABILITY METHOD

Bidu Bhusan Jena     (Faculty of Science (Mathematics), Sri Sri University, Cuttack 754006, Odisha, India)
Priyadarsini Parida     (Department of Mathematics, Kuntala Kumari Sabat Women’s College, Balasore 756003, Odisha, India)
Susanta Kumar Paikray     (Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India)

Abstract


In this paper, we prove certain Littlewood–Tauberian theorems for general matrix summability method by imposing the Tauberian conditions such as slow oscillation of usual as well as matrix generated sequence, and the De la Vallée Poussin means of real sequences. Moreover, we demonstrate \((\bar{N},p_{n})\) and \((C,1)\) – summability methods as the generalizations of our proposed general matrix method and establish an equivalence relation connecting them. Finally, we draw several remarks in view of the generalizations of some existing well-known results based on our results.


Keywords


Matrix summability, Weighted mean, Cesàro mean, Slow oscillation, Tauberian theorem

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References


  1. Ananda-Rau K. An example in the theory of summation of series by Riesz’s typical means. Proc. Lond. Math. Soc., 1930. Vol. s2-30, No. 1. P. 367–372. DOI: 10.1112/plms/s2-30.1.367
  2. Cąnak İ. A theorem on the Cesàro summability method. Comput. Math. Appl., 2011. Vol. 61, No. 4. P. 1162–1166. DOI: 10.1016/j.camwa.2010.12.065
  3. Cąnak İ. A theorem for convergence of generator sequences. Comput. Math. Appl., 2011. Vol. 61, No. 2. P. 408–411. DOI: 10.1016/j.camwa.2010.11.017
  4. Cąnak İ. An extended Tauberian theorem for the \((C,1)\) summability method. Appl. Math. Lett., 2008. Vol. 21, No. 1. P. 74–80. DOI: 10.1016/j.aml.2007.02.012
  5. Cąnak İ, Totur Ü. Some Tauberian conditions for Cesàro summability method. Math. Slovaca, 2012. Vol. 62, No. 2. P. 271–280. DOI: 10.2478/s12175-012-0008-y
  6. Cąnak İ., Totur Ü. Some Tauberian theorems for the weighted mean methods of summability. Comput. Math. Appl., 2011. Vol. 62, No. 6. P. 2609–2615. DOI: 10.1016/j.camwa.2011.07.066
  7. Cąnak İ., Totur Ü. A condition under which slow oscillation of a sequence follows from Cesàro summability of its generator sequence. Appl. Math. Comput., 2010. Vol. 216, No. 5. P. 1618–1623. DOI: 10.1016/j.amc.2010.03.017
  8. Cąnak İ., Totur Ü. A note on Tauberian theorems for regularly generated sequences. Tamkang J. Math., 2008, Vol. 39, No. 2. P. 187–191. DOI: 10.5556/j.tkjm.39.2008.29
  9. Dik M. Tauberian theorems for sequences with moderately oscillatory control modulo. Math. Moravica, 2001. Vol. 5. P. 57–94. DOI: 10.5937/MatMor0105057D
  10. Hardy G.H. Divergent Series. Oxford: Clarendon Press, 1949. 396 p.
  11. Hardy G.H., Littlewood J.E. Tauberian theorems concerning power series and Dirichlet’s series whose coefficients are positive. Proc. Lond. Math. Soc., 1914. Vol. s2-13, No. 1. P. 174–191. DOI: 10.1112/plms/s2-13.1.174
  12. Jakimovski A. On a Tauberian theorem by O. Szász. Proc. Amer. Math. Soc., 1954. Vol. 5, No. 1. P. 67–70. DOI: 10.2307/2032107
  13. Jena B.B., Paikray S.K., Misra U.K. Inclusion theorems on general convergence and statistical convergence of \((L,1,1)\)-summability using generalized Tauberian conditions. Tamsui Oxf. J. Inf. Math. Sci., 2017. Vol. 31, No. 1. P. 101–115. 
  14. Jena B.B., Paikray S.K., Misra U.K. A Tauberian theorem for double Cesàro summability method. Int. J. Math. Math. Sci., 2016. Vol. 2016. Art. no. 2431010. P. 1–4. DOI: 10.1155/2016/2431010
  15. Jena B.B., Paikray S.K., Misra U.K. A proof of Tauberian theorem for Cesàro summability method. Asian J. Math. Comput. Res., 2016. Vol. 8, No. 3. P. 272–276.
  16. Jena B.B., Paikray S.K., Parida P., Dutta H. Results on Tauberian theorem for Cesàro summable double sequences of fuzzy numbers. Kragujevac J. Math., 2020. Vol. 44, No. 4. P. 495–508. DOI: 10.46793/KgJMat2004.495J
  17. Kratz W., Stadtmüller U. Tauberian theorems for \(J_p\)-summability. J. Math. Anal. Appl., 1989. Vol. 139, No. 2. P. 362–371. DOI: 10.1016/0022-247X(89)90113-3
  18. Kratz W., Stadtmüller U. Tauberian theorems for general \(J_p\)-methods and a characterization of dominated variation. J. Lond. Math. Soc. (2), 1989. Vol. s2–39, No. 1. P. 145–159. DOI: 10.1112/jlms/s2-39.1.145
  19. Landau E. Über einen Satz des herrn Littlewood. Rend. Circ. Matem. Palermo, 1913. Vol. 35. P. 265–276. DOI: 10.1007/BF03015606 (in German)
  20. Littlewood J.E. The converse of Abel’s theorem on power series. Proc. Lond. Math. Soc., 1911. Vol. s2-9, No. 2. P. 434–448.
  21. Móricz F., Rhoades B.E. Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability. Acta Math. Hung., 1995. Vol. 66. P. 105–111. DOI: 10.1007/BF01874356
  22. Parida P., Paikray S.K., Jena B.B. Statistical Tauberian theorems for Cesàro integrability mean based on post-quantum calculus. Arab. J. Math., 2020. Vol. 9. P. 653–663. DOI: 10.1007/s40065-020-00284-z
  23. Parida P., Paikray S.K., Jena B.B. Tauberian theorems for statistical Cesàaro summability of function of two variables over a locally convex space. In: Recent Advances in Intelligent Information Systems and Applied Mathematics (ICITAM 2019), O. Castillo and al. (eds). Stud. Comput. Intell., vol 863. Cham: Springer, 2020. P. 779–790. DOI: 10.1007/978-3-030-34152-7_60
  24. Parida P., Paikray S.K., Dutta H., Jena B.B., Dash M. Tauberian theorems for Cesàro summability of \(n\)-th sequences. Filomat, 2018. Vol. 32, No. 11. P. 3993–4004. DOI: 10.2298/FIL1811993P
  25. Schmidt R. Über divergente Folgen und lineare Mittelbildungen. Math. Z., 1925. Vol. 22. P. 89–152. DOI: 10.1007/BF01479600 (in German)
  26. Szász O. On a Tauberian theorem for Abel summability. Pacific J. Math., 1951. Vol. 1, No. 1. P. 117–125.
  27. Tauber A. Ein Satz aus der Theorie der unendlichen Reihen. Monatsh. Math., 1897. Vol. 8. P. 273–277. DOI: 10.1007/BF01696278 (in German)
  28. Totur Ü., Dik M. One-sided Tauberian conditions for a general summability method. Math. Comput. Model., 2011. Vol. 54, No. 11–12. P. 2639–2644. DOI: 10.1016/j.mcm.2011.06.041
  29. Tietz H. Schmidtsche Umkehrbedingungen für Potenzreihenverfahren. Acta Sci. Math., 1990. Vol. 54. P. 355–365. (in German)
  30. Tietz H., Trautner R. Tauber Sätze für Potenzreihenverfahren. Arch. Math., 1988. Vol. 50. P. 164–174. DOI: 10.1007/BF01194575 (in German)
  31. Tietz H., Zeller K. Tauber-Bedingungen für Verfahren mit Abschnittskonvergenz. Acta Math. Hungar., 1998. Vol. 81. P. 241–247. DOI: 10.1023/A:1006598501040 (in German)




DOI: http://dx.doi.org/10.15826/umj.2024.2.008

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