Английская Википедия:Almost convergent sequence

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

Lorentz proved that <math>(x_n)</math> is almost convergent if and only if

<math>\lim\limits_{p\to\infty} \frac{x_{n}+\ldots+x_{n+p-1}}p=L</math>

uniformly in <math>n</math>.

The above limit can be rewritten in detail as

<math>\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.</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.
• Шаблон:Citation.

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