Proceedings of the American Mathematical Society

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

 

 

Statistical extensions
of some classical Tauberian theorems


Authors: J. A. Fridy and M. K. Khan
Journal: Proc. Amer. Math. Soc. 128 (2000), 2347-2355
MSC (1991): Primary 40E05
Published electronically: February 25, 2000
MathSciNet review: 1653457
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence $x$ satisfies $\lim Cx = L$ and $\Delta x_k = O (1/k)$, then $\lim x = L$. In this paper it is shown that the hypothesis $\lim Cx = L$ can be replaced by the weaker assumption of the statistical limit: st-lim $Cx = L$, i.e., for every $\epsilon >0$, $\lim n^{-1} | \{ k \leq n: | ( Cx)_k - L | \geq \epsilon \} | = 0$. Similarly, the ``one-sided'' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing $\lim Cx $ and $\lim Ax$ with st-lim $Cx$ and st-lim $Ax$, respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing $\lim _t (Bx)_t=L$, where $t$ is a continuous parameter, with $\lim _n (Bx)_n =L$, and further replacing it by $(B^{*})$-st-lim $B^{*} x =L$, where $B^{*}$ is the Borel matrix method.


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

  • 1. David H. Armitage and Ivor J. Maddox, Discrete Abel means, Analysis 10 (1990), no. 2-3, 177–186. MR 1074831
  • 2. N. H. Bingham, Tauberian theorems and the central limit theorem, Ann. Probab. 9 (1981), no. 2, 221–231. MR 606985
  • 3. Donato Greco, Criteri di compattezza per insiemi di funzioni in 𝑛 variabili indipendenti, Ricerche Mat. 1 (1952), 124–144 (Italian). MR 0048550
    C. J. Pipes, Generalizations of a theorem of Sierpinski and Zygmund on continuous functions, Proc. Amer. Math. Soc. 3 (1952), 237–243. MR 0048549, 10.1090/S0002-9939-1952-0048549-8
    Charles A. Hayes, Differentiation of some classes of set functions, Proc. Cambridge Philos. Soc. 48 (1952), 374–382. MR 0048546
    H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241–244 (1952) (French). MR 0048548
    L. Pukánszky and A. Rényi, On the approximation of measurable functions, Publ. Math. Debrecen 2 (1951), 146–149. MR 0048547
  • 4. J. A. Fridy and M. K. Khan, Tauberian theorems via statistical convergence, J. Math. Anal. Appl. 228 (1998), no. 1, 73-95. CMP 99:05
  • 5. J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. MR 816582, 10.1524/anly.1985.5.4.301
  • 6. G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc. 8 (1910), no. 2, 310-320.
  • 7. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • 8. G. H. Hardy and J. E. Littlewood, Theorems concerning the summability of series by Borel's exponential method, Rend. Circ. Mat. Palermo 41 (1910), no. 2, 36-53.
  • 9. -, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, P. Lond. Math. Soc. 13 (1914), 174-191.
  • 10. K. Knopp, Über das Eulershe Summierungsverfahren, Math. Zeit. 18 (1923), no. II, 125-156.
  • 11. E. Landau, Über die Bedentung einiger Grenzwertsätze der Herren Hardy und Axer, Prace Mat.-fiz. 21 (1910), 97-177.
  • 12. J. E. Littlewood, The converse of Abel's theorem on power series, P. Lond. Math. Soc. 9 (1910), no. 2, 434-448.
  • 13. R. E. Powell and S. M. Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972.
  • 14. R. Schmidt, Über divergente Folgen und Mittelbildungen, M. Zeit. 22 (1925), 89-152.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 40E05

Retrieve articles in all journals with MSC (1991): 40E05


Additional Information

J. A. Fridy
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Email: fridy@mcs.kent.edu

M. K. Khan
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Email: kazim@mcs.kent.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05241-2
Keywords: Statistical convergence, Tauberian theorems
Received by editor(s): March 5, 1998
Received by editor(s) in revised form: September 17, 1998
Published electronically: February 25, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society