# σ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş; Richard Patterson

Open Mathematics (2006)

- Volume: 4, Issue: 4, page 648-655
- ISSN: 2391-5455

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topEkrem Savaş, and Richard Patterson. "σ-asymptotically lacunary statistical equivalent sequences." Open Mathematics 4.4 (2006): 648-655. <http://eudml.org/doc/269459>.

@article{EkremSavaş2006,

abstract = {This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 \[\mathop \{\lim \}\limits \_r \frac\{1\}\{\{h\_r \}\}\left\lbrace \{k \in I\_r :\left| \{\frac\{\{x\_\{\sigma ^k (m)\} \}\}\{\{y\_\{\sigma ^k (m)\} \}\} - L\} \right| \geqslant \in \} \right\rbrace = 0\]
uniformly in m = 1, 2, 3, ..., (denoted by x \[\mathop \sim \limits ^\{S\_\{\sigma ,\theta \} \} \]
y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented.},

author = {Ekrem Savaş, Richard Patterson},

journal = {Open Mathematics},

keywords = {40A99; 40A05},

language = {eng},

number = {4},

pages = {648-655},

title = {σ-asymptotically lacunary statistical equivalent sequences},

url = {http://eudml.org/doc/269459},

volume = {4},

year = {2006},

}

TY - JOUR

AU - Ekrem Savaş

AU - Richard Patterson

TI - σ-asymptotically lacunary statistical equivalent sequences

JO - Open Mathematics

PY - 2006

VL - 4

IS - 4

SP - 648

EP - 655

AB - This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 \[\mathop {\lim }\limits _r \frac{1}{{h_r }}\left\lbrace {k \in I_r :\left| {\frac{{x_{\sigma ^k (m)} }}{{y_{\sigma ^k (m)} }} - L} \right| \geqslant \in } \right\rbrace = 0\]
uniformly in m = 1, 2, 3, ..., (denoted by x \[\mathop \sim \limits ^{S_{\sigma ,\theta } } \]
y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented.

LA - eng

KW - 40A99; 40A05

UR - http://eudml.org/doc/269459

ER -

## References

top- [1] G. Das and B.K. Patel: “Lacunary distribution of sequences”, Indian J. Pure Appl. Math., Vol. 26(1), (1989), pp. 54–74. Zbl0726.40002
- [2] H. Fast: “Sur la convergence statistique”, Collog. Math., Vol. 2, (1951), pp. 241–244. Zbl0044.33605
- [3] J.A. Fridy: “Minimal rates of summability”, Can. J. Math., Vol. 30(4), (1978), pp. 808–816. Zbl0359.40003
- [4] J.A. Fridy: “On statistical sonvergence”, Analysis, Vol. 5, (1985), pp. 301–313. Zbl0588.40001
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- [6] G.G. Lorentz: “A contribution to the theory of divergent sequences”, Acta. Math., Vol. 80, (1948), pp. 167–190. http://dx.doi.org/10.1007/BF02393648 Zbl0031.29501
- [7] Mursaleen: “Some new spaces of lacunary sequences and invariant means”, Ital. J. Pure Appl. Math., Vol. 11, (2002), pp. 175–181. Zbl1104.46003
- [8] Mursaleen: “New invariant matrix methods of summability”, Quart. J. Math. Oxford, Vol. 34(2), (1983), pp. 133, 77–86.
- [9] M. Marouf: “Asymptotic equivalence and summability”, Int. J. Math. Math. Sci., Vol. 16(4), (1993), pp. 755–762. http://dx.doi.org/10.1155/S0161171293000948 Zbl0788.40001
- [10] R.F. Patterson: “On asymptotically statistically equivalent sequences”, Demonstratio Math., Vol. 36(1), (2003), pp. 149–153. Zbl1045.40003
- [11] R.F. Patterson and E. Savaş: “On asymptotically lacunary statistically equivalent sequences”, (in press). Zbl1155.40301
- [12] P. Schaefer: “Infinite matrices and invariant means”, Proc. Amer. Math. Soc., Vol. 36, (1972), pp. 104–110. http://dx.doi.org/10.2307/2039044 Zbl0255.40003

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