Tauberian conditions under which convergence follows from summability by the discrete power series method

Tauberian conditions under which convergence follows from summability by the discrete power series method

In this paper, we obtain Tauberian conditions to recover convergence of a series from its discrete power seriessummability under certain conditions. As special cases of our main results, we get discrete analogues of some well-knownTauberian theorems in the literature.

___

  • [1] Armitage DH, Maddox IJ. Discrete Abel means. Analysis. International Mathematical Journal of Analysis and its Applications 1990; 10 (2-3), 177-186. doi: 10.1524/anly.1990.10.23.177
  • [2] Çanak İ, Totur Ü. Tauberian theorems for the (J, p) summability method. Applied Mathematics Letters 2012; 25 (10): 1430-1434. doi: 10.1016/j.aml.2011.12.017
  • [3] Çanak İ, Totur Ü. A Tauberian theorem for the discrete Mφ summability method. Applied Mathematics Letters 2012; 25 (4): 771-774. doi: 10.1016/j.aml.2011.10.018
  • [4] Çanak İ, Totur Ü. A theorem for the (J, p) summability method. Acta Mathematica Hungarica 2015; 145 (1): 220-228. doi: 10.1007/s10474-014-0452-y
  • [5] Hardy GH. Divergent Series. Oxford, UK: Clarendon Press, 1949.
  • [6] Ishiguro K. A Tauberian theorem for (J, pn) summability. Proceedings of the Japan Academy 1964; 40: 807-812. doi: 10.3792/pja/1195522569
  • [7] Ishiguro K. Two Tauberian theorems for (J, pn) summability. Proceedings of the Japan Academy 1965; 41: 40-45. doi: 10.3792/pja/1195522526
  • [8] Korevaar J. Tauberian Theory: A Century of Developments. Berlin, Germany: Springer-Verlag, 2004.
  • [9] Maddox IJ. Tauberian theorems for some classes of discrete Abel means. Radovi Matematički 1990; 6 (2): 273-278.
  • [10] Maddox IJ. A Tauberian theorem for discrete Abel means. Indian Journal of Mathematics 1991; 33 (1): 7-10.
  • [11] Patterson RF, Sen P, Rhoades BE. A Tauberian theorem for a generalized power series method. Applied Mathematics Letters 2005; 18 (10): 1129-1133. doi: 10.1016/j.aml.2004.11.006
  • [12] Sezer SA, Çanak İ. Conditions for the equivalence of power series and discrete power series methods of summability. Filomat 2015; 29 (10): 2275-2280. doi: 10.2298/FIL1510275S
  • [13] Sezer SA, Çanak İ. Power series methods of summability for series of fuzzy numbers and related Tauberian theorems. Soft Computing 2017; 21 (4): 1057-1064. doi: 10.1007/s00500-015-1840-0
  • [14] Sezer SA, Canak I. On converse theorems for the discrete Bürmann power series method of summability. Maejo International Journal of Science and Technology 2016; 10 (3): 346-353.
  • [15] Tauber A. Ein Satz aus der Theorie der unendlichen Reihen. Monatshefte für Mathematik und Physik 1897; 8: 273-277 (in German). doi: 10.1007/BF01696278
  • [16] Totur Ü, Çanak İ. A Tauberian theorem for the power series summability method. Ukrainian Mathematical Journal 2017; 69 (12): 1701-1713. doi: 10.1007/s11253-018-1482-3
  • [17] Watson B. Discrete power series methods. Analysis. International Mathematical Journal of Analysis and its Applications 1998; 18 (1): 97-102. doi: 10.1524/anly.1998.18.1.97
  • [18] Watson B. A Tauberian theorem for discrete power series methods. Analysis. International Mathematical Journal of Analysis and its Applications 2002; 22 (4): 361-365. doi: 10.1524/anly.2002.22.4.361