Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

Abstract

The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which , where and denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.

On Asymptotically f-statistical Equivalent Set Sequences in the Sense of Wijsman

“…In the year 2014, Aizpuru et al [1] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of non-matrix convergence, namely, f -statistical convergence, which is intermediate between the ordinary convergence and the statistical convergence and agrees with the statistical convergence when the modulus function is the identity mapping. Quite recently, Bhardwaj and Dhawan [3], and Bhardwaj et al [4], have introduced and studied the concepts of f -statistical convergence of order α and f -statistical boundedness, respectively, by using the approach of Aizpuru et al [1] (see also [5][6][7][8]).…”

“…Similar to this concept, the concept of Wijsman lacunary statistical convergence was presented by Ulusu and Nuray [25] in 2012. For further results one may refer to [6,10,[15][16][17][24][25][26][27][28][29]. The notion of Wijsman statistical convergence has been extended by Bhardwaj et al [7] to that of f -Wijsman statistical convergence, where f is an unbounded modulus.…”

Application of f-lacunary statistical convergence to approximation theorems

“…For example, the modulus where , is unbounded, but is bounded. For the work related to sequence spaces defined by a modulus, one may refer to [ 1 , 5 – 7 , 9 , 10 , 25 ] and many others.…”