Abstract
In this paper, we introduce the concepts of lacunary statistical τ -convergence, lacunary statistically τ -bounded and lacunary statistically τ -Cauchy in the framework of locally solid Riesz spaces. We also define a new type of convergence, that is, S ∗ (τ )-convergence in this setup and prove some interesting results related to these notions. MSC: 40A35; 40G15; 46A40
Highlights
Introduction and preliminaries InFast [ ] presented the following definition of statistical convergence for sequences of real numbers
The natural density of K is defined by δ(K) = limn n– |Kn| if the limit exists, where the vertical bars indicate the number of elements in the enclosed set
We say that a sequence x = in X is lacunary statistically τ -bounded (or Sθ (τ )-bounded) if for every τ -neighborhood U of zero there exists some λ > such that the set MU = {j ∈ N : λxj ∈/ U} has θ -density zero (shortly, δθ (MU ) = ), i.e
Summary
Introduction and preliminaries InFast [ ] presented the following definition of statistical convergence for sequences of real numbers. The sequence x = (xk) is said to be statistically convergent to L if for every ε > , the set Kε := {k ∈ N : |xk – L| ≥ ε} has natural density zero (cf Fast [ ]), i.e., for each ε > , A linear topology τ on a Riesz space X is said to be locally solid [ ] if τ has a base at zero consisting of solid sets.
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