Lacunary statistical boundedness on time scales

Bayram Sözbir,Selma Altundağ

doi:10.1186/s13662-021-03656-7

Abstract

In this paper, we introduce the concept of lacunary statistical boundedness of Δ-measurable real-valued functions on an arbitrary time scale. We also give the relations between statistical boundedness and lacunary statistical boundedness on time scales.

Highlights

The idea of statistical convergence was formally introduced by Fast [1] and Steinhaus [2], independently
The natural density of K is defined by δ(K) = limnn–1|Kn| if the limit exists, where |Kn| indicates the cardinality of Kn
The idea of statistical convergence was first studied on time scales in [26] and [27], independently

Summary

Introduction

The idea of statistical convergence was formally introduced by Fast [1] and Steinhaus [2], independently. Definition 1.1 ([27]) A -measurable function f : T → R is statistically convergent to a number L on T if, for every ε > 0, lim μ ({s ∈ [t0, t]T : |f (s) – L| ≥ ε}) = 0, t→∞ A -measurable function f : T → R is said to be lacunary statistically convergent to a number L if, for every ε > 0, lim μ ({s ∈ (kr–1, kr]T : |f (s) – L| ≥ ε}) = 0, r→∞

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Conclusion