Almost convergent sequence

A bounded real sequence [math]\displaystyle{ (x_n) }[/math] is said to be almost convergent to [math]\displaystyle{ L }[/math] if each Banach limit assigns the same value [math]\displaystyle{ L }[/math] to the sequence [math]\displaystyle{ (x_n) }[/math].

Lorentz proved that [math]\displaystyle{ (x_n) }[/math] is almost convergent if and only if

[math]\displaystyle{ \lim\limits_{p\to\infty} \frac{x_{n}+\ldots+x_{n+p-1}}p=L }[/math]

uniformly in [math]\displaystyle{ n }[/math].

The above limit can be rewritten in detail as

[math]\displaystyle{ \forall \varepsilon\gt 0 : \exists p_0 : \forall p\gt p_0 : \forall n : \left|\frac{x_{n}+\ldots+x_{n+p-1}}p-L\right|\lt \varepsilon. }[/math]

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, https://archive.org/details/divergentseries033523mbp .

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