Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cesàro and Borel-type summability

Authors: David Borwein and Tom Markovich
Journal: Proc. Amer. Math. Soc. 103 (1988), 1108-1112
MSC: Primary 40G05; Secondary 40E05, 40G10
MathSciNet review: 954991
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Though summability of a series by the Cesàro method $ {C_p}$ does not in general imply its summability by the Borel-type method $ (B,\alpha ,\beta )$, it is shown that the implication holds under an additional condition.

References [Enhancements On Off] (What's this?)

  • [1] D. Borwein, A Tauberian theorem for Borel-type methods of summability, Canad. J. Math. 21 (1969), 740–747. MR 0243233
  • [2] D. Borwein and T. Markovich, A Tauberian theorem concerning Borel-type and Cesàro methods of summability, (submitted).
  • [3] Gerhard Faulhaber, Äquivalenzsätze für die Kreisverfahren der Limitierungstheorie, Math. Z. 66 (1956), 34–52 (German). MR 0082572
  • [4] G. H. Hardy and J. E. Littlewood, Theorems concerning the summability of series by Borel's exponential method, Rend. Circ. Mat. Palermo 41 (1916), 36-53.
  • [5] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • [6] J. M. Hyslop, Note on certain related conditions in the theory of Cesàro summability, Proc. Edinburgh Math. Soc. (2) 6 (1940), 166–171. MR 0002638
  • [7] -, The generalisation of a theorem on Borel summability, Proc. London Math. Soc. (2) 41 (1936), 243-256.
  • [8] B. Kwee, An improvement on a theorem of Hardy and Littlewood, J. London Math. Soc. (2) 28 (1983), no. 1, 93–102. MR 703468, 10.1112/jlms/s2-28.1.93
  • [9] Amram Meir, Tauberian constants for a family of transformations, Ann. of Math. (2) 78 (1963), 594–599. MR 0166519
  • [10] V. Swaminathan, A note on the family 𝐹(𝑎,𝑞) of summability methods, Math. Z. 138 (1974), 119–122. MR 0348330

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40G05, 40E05, 40G10

Retrieve articles in all journals with MSC: 40G05, 40E05, 40G10

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society