Lacunary statistical convergence of double sequences

We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong (A, φ) -convergence with respect to a modulus function. In addition, we also study some relationships between double lacunary strong (A, φ) -convergence with respect to a modulus and double lacunary statistical convergence.

[1]  Alfred Pringsheim,et al.  Zur Theorie der zweifach unendlichen Zahlenfolgen , 1900 .

[2]  G. M. Robison.  Divergent double sequences and series , 1926 .

[3]  J. A. Fridy,et al.  Lacunary statistical convergence. , 1993 .

[4]  William H. Ruckle,et al.  FK Spaces in Which the Sequence of Coordinate Vectors is Bounded , 1973, Canadian Journal of Mathematics.

[5]  R. Patterson.  ANALOGUES OF SOME FUNDAMENTAL THEOREMS OF SUMMABILITY THEORY , 2000 .

[6]  Jinlu Li.  Lacunary statistical convergence and inclusion properties between lacunary methods , 2000 .

[7]  Allen R. Freedman,et al.  SOME CESARO-TYPE SUMMABILITY SPACES , 1978 .

[8]  J. S. Connor,et al.  THE STATISTICAL AND STRONG p-CESARO CONVERGENCE OF SEQUENCES , 1988 .

[9]  Jeff Connor,et al.  On Strong Matrix Summability with Respect to a Modulus and Statistical Convergence , 1989, Canadian Mathematical Bulletin.

[10]  Osama H. H. Edely,et al.  Statistical Convergence of Double Sequences , 2003 .

[11]  A. M. Brono,et al.  Lacunary Statistical Convergence of Double Sequences of Order α of Sets 1 , 2019 .

[12]  Ekrem Savas,et al.  Erratum to "Lacunary statistical convergence of multiple sequences" [Appl. Math. Lett. 19(2006) 527-534] , 2007, Appl. Math. Lett..

[13]  Ekrem Savas,et al.  RH-conservative matrix characterization of P-convergence in probability , 2012, Comput. Math. Appl..

[14]  H. J. Hamilton.  Transformations of multiple sequences , 1936 .

[15]  I. J. Maddox.  On strong almost convergence , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  H. Fast,et al.  Sur la convergence statistique , 1951 .

[17]  Ekrem Savas,et al.  Consistent classes of double summability methods , 2010, Appl. Math. Lett..

[18]  E. Savaş.  A NOTE ON DOUBLE SEQUENCES OF FUZZY NUMBERS , 1996 .

[19]  E. Savaş,et al.  Matrix summability of statistically p-convergence sequences , 2011 .

[20]  I. J. Maddox.  Sequence spaces defined by a modulus , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[22]  E. Savaş,et al.  Some σ-double sequence spaces defined by Orlicz function , 2006 .

[23]  Ekrem Savas,et al.  Lacunary statistical convergence of multiple sequences , 2006, Appl. Math. Lett..

[24]  J. A. Fridy,et al.  Lacunary Statistical Summability , 1993 .