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

DOI:
https://doi.org/10.1090/S0002-9939-00-05241-2

Published electronically:
February 25, 2000

MathSciNet review:
1653457

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided'' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

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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:
https://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