On \(\sigma\)-statistically convergence and lacunary \(\sigma\)-statistically convergence. (English) Zbl 0794.40001
Using the well-known definition of statistical convergence, the authors introduce the \(\sigma\)-statistical convergence and lacunary \(\sigma\)- statistical convergence. Then they study the inclusion relations between these two new types of convergence and shows that these two types of convergence are equivalent for bounded sequences.
Reviewer: D.Somasundaram (Salem)
MSC:
| 40A05 | Convergence and divergence of series and sequences |
| 40C05 | Matrix methods for summability |
| 40D05 | General theorems on summability |
Keywords:
statistical convergenceReferences:
| [1] | CONNOR J.: The statistical and strong p-Cesáro convergence of sequences. Analysis 8 (1988), 47-63. · Zbl 0653.40001 |
| [2] | FAST H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 |
| [3] | FREEDMAN A., SEMBER J., RAPHAEL M.: Some Cesáro-type summability spaces. Proc. London Math. Soc. (3) 37 (1978), 508-520. · Zbl 0424.40008 · doi:10.1112/plms/s3-37.3.508 |
| [4] | FRIDY J.: On statistical convergence. Analysis 5 (1985), 301-310. · Zbl 0588.40001 |
| [5] | LORENTZ G. G.: A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167-190. · Zbl 0031.29501 · doi:10.1007/BF02393648 |
| [6] | MADDOX I. J.: Statistical convergence in a locally convex space. Math. Proc. Cambridge Philos. Soc. 104 (1988), 141-145. · Zbl 0674.40008 · doi:10.1017/S0305004100065312 |
| [7] | MURSALEEN: Invariant means and some matrix transformations. Tamkang J. Math. 10 (1979), 183-188. · Zbl 0437.40004 |
| [8] | MURSALEEN: Matrix transformations between some new sequence spaces. Houston J. Math. 9 (1983), 505-509. · Zbl 0542.40003 |
| [9] | ŠALÁT T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003 |
| [10] | SAVAŞ E.: On lacunary strong \sigma -convergence. Indian J. Pure Appl. Math. 21 (1990), 359-365. · Zbl 0715.40001 |
| [11] | SAVAŞ E.: Some sequence spaces involving invariant means. Indian J. Math. 31 (1989), 1-8. · Zbl 0681.46008 |
| [12] | SCHAEFER P.: Infinite matrices and invariant means. Proc. Amer. Math. Soc. 36 (1972), 104-110. · Zbl 0255.40003 · doi:10.2307/2039044 |
| [13] | SCHOENBERG I. J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361-365. · Zbl 0089.04002 · doi:10.2307/2308747 |
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