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On the invariant mean and statistical convergence

Authors
Journal
Applied Mathematics Letters
0893-9659
Publisher
Elsevier
Publication Date
Volume
22
Issue
11
Identifiers
DOI: 10.1016/j.aml.2009.06.005
Keywords
  • Banach Limit
  • Invariant Mean
  • Almost Convergence
  • [Formula Omitted]-Convergence
  • [Formula Omitted]-Density
  • Statistical Convergence

Abstract

Abstract Two concepts - one of almost convergence and the other of statistical convergence - play a very active role in recent research on summability theory. The definition of almost convergence introduced by Lorentz [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190] originated from the concept of the Banach limit, while the statistical convergence introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] was defined through the concept of density. Both involve non-matrix methods of summability and they are incompatible. In this work we define two new kinds of summability methods by using these two mutually incompatible concepts of the Banach limit and of density to deal with those sequences which are statistically convergent but not almost convergent or vice versa.

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