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Zbl 0633.40001
Butković, D.; Kraljević, H.; Sarapa, N.
On the almost convergence. (English)
[A] Functional analysis II, Proc. 2nd Conf., Dubrovnik/Yugosl. 1985, Lect. Notes Math. 1242, 396-417 (1987).

[For the entire collection see Zbl 0615.00007.] \par Let $(x\sb n)$ be a real or complex sequence, $\omega$ : ${\bbfN}\to {\bbfN}\sp a $strictly increasing function, and $(x\sp{\omega}\sb n)$ a sequence defined by $x\sp{\omega}\sb n=x\sb k$ if $\omega (k)=n$ for some $k\in {\bbfN}$ and $x\sp{\omega}\sb n=0$ otherwise. Let $\phi\sb n$ be 1 if $n=\omega (k)$ for some $k\in {\bbfN}$ and $\phi\sb n=0$ otherwise. The authors prove that if $(x\sb n)$ and $(\phi\sb n)$ are almost convergent, $(x\sp{\omega}\sb n)$ is also convergent and the Cesàro limit Lim $x\sp{\omega}\sb n$ equals Lim $\phi$ ${}\sb n Lim x\sb n$. This and similar results lead to generalizations of some results of {\it B. E. Rhoades} [Approximation Theory III, Proc. Conf. Hon. G. G. Lorentz, Austin/Tex. 1980, 735-740 (1980; Zbl 0476.60061)] and of some formulas of {\it A. T. Bharucha-Reid} [Stud. Math. 17, 189-197 (1958; Zbl 0101.095)] with applications in the theory of Markov chains.

MSC 2000:: *40A05 Convergence of series and sequences
40C05 Matrix methods in summability
60J05 Markov processes with discrete parameter

Keywords: dilution functions; strongly regular matrices; Cesàro limit; Markov chains

Citations: Zbl 0615.00007; Zbl 0476.60061; Zbl 0101.095

Cited in: Zbl 0830.46005 Zbl 0732.40006 Zbl 0633.40002

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