σ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş; Richard Patterson

Open Mathematics (2006)

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

Abstract

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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 lim r 1 h r k ∈ I r : x σ k ( m ) y σ k ( m ) - L ⩾ ∈ = 0 uniformly in m = 1, 2, 3, ..., (denoted by x ∼ S σ , θ 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.

How to cite

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Ekrem 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

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  1. [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. [2] H. Fast: “Sur la convergence statistique”, Collog. Math., Vol. 2, (1951), pp. 241–244. Zbl0044.33605
  3. [3] J.A. Fridy: “Minimal rates of summability”, Can. J. Math., Vol. 30(4), (1978), pp. 808–816. Zbl0359.40003
  4. [4] J.A. Fridy: “On statistical sonvergence”, Analysis, Vol. 5, (1985), pp. 301–313. Zbl0588.40001
  5. [5] J.A. Fridy and C. Orhan: “Lacunary statistical sonvergent”, Pacific J. Math., Vol. 160(1), (1993), pp. 43–51. Zbl0794.60012
  6. [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. [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. [8] Mursaleen: “New invariant matrix methods of summability”, Quart. J. Math. Oxford, Vol. 34(2), (1983), pp. 133, 77–86. 
  9. [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. [10] R.F. Patterson: “On asymptotically statistically equivalent sequences”, Demonstratio Math., Vol. 36(1), (2003), pp. 149–153. Zbl1045.40003
  11. [11] R.F. Patterson and E. Savaş: “On asymptotically lacunary statistically equivalent sequences”, (in press). Zbl1155.40301
  12. [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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