Self-phase modulation: Difference between revisions

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A [[bounded sequence|bounded]] [[real number|real]] [[sequence]] <math>(x_n)</math> is said to be ''almost convergent'' to <math>L</math> if each [[Banach limit]] assigns
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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 [[Nigel Kalton|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.
 
{{PlanetMath attribution|id=7356|title=Almost convergent}}
 
[[Category:Convergence (mathematics)]]
[[Category:Sequences and series]]

Latest revision as of 11:23, 2 July 2014

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