Further remarks on uniform statistical convergence of order $\alpha$

Abstract

In this note, following recent works [Colak (Statistical convergence of order α, Modern methods in Analysis and its, pp 121–129, 2010), Colak and Bektas (Acta Mathematica Scientia 31B(3): 953–959, 2011), Das et al. (Acta Mathematica Hungarica 134 (1–2):153–161, 2012)], we make a new approach to a well known summability method corresponding to uniform density [Balaz and Salat (Mathematical Communications 11: 1–7, 2006)] which is also called uniform statistical convergence [Albayrak and Pehlivan (Applied Mathematics Letters, 2010)] and introduce a new notion called $I_u-$convergence of order $\alpha$ (or uniform statistical convergence of order $\alpha$) and establish its basic properties and try to understand how this new approach affects the behaviors of the known summability method.