A Contribution to the Theory of Divergent Sequences

Abstract

In this paper we define and examine a new method of summation which x This assigns a general limit Lim x,~ to certain bounded sequences x-~ xx~j. method is analogous to the mean values which are used in the theory of almost periodic functions, furthermore it is narrowly connected with the limits of S.

On σ-bounded variation and ideal convergence of fuzzy sequences

“…In case σ is translation mapping n → n + 1, a σmean is often called a Banach limit (see [3]) and V σ , the set of bounded sequences all of whose invariant means are equal, if σ(n) = n + 1, the it is called the set of almost convergent sequences (see [26]). …”

Singular symmetric functionals and Banach limits with additional invariance properties

“…These functionals are called Banach limits, and the method itself extends the original result of Dixmier [1], who used generalized limits invariant under the operation of dilation. Here we note that to justify our construction we have made essential use of the results in [6] and [5] on the theory of Banach limits.…”

“…For a detailed study of Banach limits in BL(l ∞ ), see [5] and [6]. Lorentz [5] introduced and studied the notion of an almost convergent sequence. A sequence {x n } is said to be almost convergent to a limit a if the relation L({x n }) = a holds for all Banach limits L in BL(l ∞ ).…”

“…is almost convergent to 1/2. Theorem 1.2 [5]. For a sequence {x n } to be almost convergent to a it is necessary and sufficient that the limiting relation…”

“…Following the terminology of Lorentz [5], we shall call the generalized limit of an almost convergent sequence {x n } the F -limit and denote it by F -lim n x n .…”