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Almost convergent sequence

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A bounded real sequence ( x n ) {\displaystyle (x_{n})} is said to be almost convergent to L {\displaystyle L} if each Banach limit assigns the same value L {\displaystyle L} to the sequence ( x n ) {\displaystyle (x_{n})} .

Lorentz proved that ( x n ) {\displaystyle (x_{n})} is almost convergent if and only if

lim p → ∞ x n + … + x n + p − 1 p = L {\displaystyle \lim \limits _{p\to \infty }{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}=L}

uniformly in n {\displaystyle n} .

The above limit can be rewritten in detail as

∀ ε > 0 : ∃ p 0 : ∀ p > p 0 : ∀ n : | x n + … + x n + p − 1 p − L | < ε . {\displaystyle \forall \varepsilon >0:\exists p_{0}:\forall p>p_{0}:\forall n:\left|{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}-L\right|<\varepsilon .}

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.
  • Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press.
Specific
  1. Hardy, p. 52

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