Abstract
By using modulus functions, we have obtained a generalization of statistical convergence of asymptotically equivalent sequences, a new non-matrix convergence method, which is intermediate between the ordinary convergence and the statistical convergence. Further, we have examined some inclusion relations related to this concept.
Highlights
In 1993, Marouf [8] presented definitions for asymptotically equivalent and asymptotic regular matrices
In 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 nonmatrix 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
We have defined a generalization of statistical convergence of asymptotically equivalent sequences and obtained some inclusion relations related to this concept
Summary
In 1993, Marouf [8] presented definitions for asymptotically equivalent and asymptotic regular matrices. In 2003, Patterson [11] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and. In 2006, Patterson and Savas extended the definitions presented in [12] to lacunary sequences. Ruckle [13] and Maddox [7] introduced and discussed some properties of sequence spaces defined by using a modulus function. Bhardwaj and Dhawan [2], and Bhardwaj et al [3], 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 [4] and [5]). We have defined a generalization of statistical convergence of asymptotically equivalent sequences and obtained some inclusion relations related to this concept
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