Definitions

convergent-sequence

Almost convergent sequence

A bounded real sequence (x_n) is said to be almost convergent to L if each Banach limit assigns the same value L to the sequence (x_n).

Lorentz proved that (x_n) is almost convergent if and only if

limlimits_{ptoinfty} frac{x_{n}+ldots+x_{n+p-1}}p=L
uniformly in n.

The above limit can be rewritten in detail as

(forall varepsilon>0) (exists p_0) (forall p>p_0) (forall n) left|frac{x_{n}+ldots+x_{n+p-1}}p-Lright|
Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.

References

  • G. Bennett and N.J. Kalton: "Consistency theorems for almost convergence." Trans. Amer. Math. Soc., 198:23--43, 1974.
  • J. Boos: "Classical and modern methods in summability." Oxford University Press, New York, 2000.
  • J. Connor and K.-G. Grosse-Erdmann: "Sequential definitions of continuity for real functions." Rocky Mt. J. Math., 33(1):93--121, 2003.
  • G.G. Lorentz: "A contribution to the theory of divergent sequences." Acta Math., 80:167--190, 1948.

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