almost convergent

A real sequence $(x_{n})$ is said to be almost convergent to $L$ if each Banach limit assignsthe 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 methodwhich 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.