Almost convergent sequence

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

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

${\displaystyle \lim \limits _{p\to \infty }{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}=L}$

uniformly in ${\displaystyle n}$.

The above limit can be rewritten in detail as

${\displaystyle \forall \varepsilon >0:\exists p_{0}:\forall p>p_{0}:\forall n:\left|{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}-L\right|<\varepsilon .}$

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.^[1]

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

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