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
$$\underset{p\to \mathrm{\infty}}{lim}\frac{{x}_{n}+\mathrm{\dots}+{x}_{n+p-1}}{p}=L$$ |
uniformly in $n$.
The above limit can be rewritten in detail as
$$ |
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.
Title | almost convergent |
---|---|
Canonical name | AlmostConvergent |
Date of creation | 2013-03-22 15:29:51 |
Last modified on | 2013-03-22 15:29:51 |
Owner | kompik (10588) |
Last modified by | kompik (10588) |
Numerical id | 12 |
Author | kompik (10588) |
Entry type | Definition |
Classification | msc 40A05 |
Classification | msc 40C99 |
Related topic | Banachlimit |
Defines | almost convergent |