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$\lambda $-statistical convergence. (English) Zbl 0953.40002
This paper deals with a generalization of the Cesàro mean summability of the sequences $$ [C,1]:=\{x=(x_n): \text{ there is } L \in \bbfR\text{ such that } \lim _{n \to \infty }{1 \over n} \sum _{k=1}^n {}x_k - L{}=0 \}$$ to $$[V,\lambda ]:= \{x=(x_n): \text{ there is } L \in \bbfR \text{ such that } \lim_{n \to \infty }{1 \over \lambda _n} \sum _{k \in I_n} {}x_k - L{}=0{}\}$$ for some interval $I_n$. Comparisons of statistical convergence and $\lambda $-statistical convergence for a sequence $x=(x_n)$ defined by using limits $$\lim _{n \to \infty }{1 \over n} {}\{ k \leq n : {}x_n - L{} \geq \varepsilon \}=0 \quad \text{and}\quad\lim _{n \to \infty }{1 \over \lambda _n} {}\{k \in I_n : {}x_n - L{} \geq \varepsilon \}= 0$$ are given.
Reviewer: Ondrej Kováčik (Žilina)

MSC:
40A05Convergence and divergence of series and sequences
40C05Matrix methods in summability
WorldCat.org
Full Text: EuDML
References:
[1] CONNOR J. S.: The statistical and strong p-Cesáro convergence of sequences. Analysis 8 (1988), 47-63. · Zbl 0653.40001
[2] CONNOR J. S.: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 32 (1989), 194-198. · Zbl 0693.40007 · doi:10.4153/CMB-1989-029-3
[3] FAST H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 · eudml:209960
[4] FREEDMAN A. R.-SEMBER J. J.-RAPHAEL M.: Some Cesáro type summability spaces. Proc. London Math. Soc. 37 (1978), 508-520. · Zbl 0424.40008 · doi:10.1112/plms/s3-37.3.508
[5] FRIDY J. A.: On statistical convergence. Analysis 5 (1985), 301-313. · Zbl 0588.40001
[6] FRIDY J. A.-ORHAN C.: Lacunary statistical convergence. Pacific. J. Math. 160 (1993), 43-51. · Zbl 0794.60012 · doi:10.2140/pjm.1993.160.43
[7] KOLK E.: The statistical convergence in Banach spaces. Acta Comment. Univ. Tartu 928 (1991), 41-52.
[8] LEINDLER L.: Über die de la Vallée-Pousinsche Summierbarkeit allgemeiner Orthogonalreihen. Acta Math. Acad. Sci. Hungar. 16 (1965), 375-387. · Zbl 0138.28802 · doi:10.1007/BF01904844
[9] ŠALÁT T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003 · eudml:34081