almost convergent

A 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 [4] proved that $(x_{n})$ is almost convergent to $L$ if and only if

\lim\limits_{p\to\infty}\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}-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

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