On statistical A $\mathfrak{A}$ -Cauchy and statistical A $\mathfrak{A}$ -summability via ideal

Abstract

Abstract The notion of statistical convergence was extended to I $\mathfrak{I}$ -convergence by (Kostyrko et al. in Real Anal. Exch. 26(2):669–686, 2000). In this paper we use such technique and introduce the notion of statistically A I $\mathfrak{A}^{\mathfrak{I}}$ -Cauchy and statistically A I ∗ $\mathfrak{A}^{\mathfrak{I}^{\ast }}$ -Cauchy summability via the notion of ideal. We obtain some relations between them and prove that under certain conditions statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -Cauchy and statistical A I ∗ $\mathfrak{A}^{\mathfrak{I}^{\ast }}$ -Cauchy summability are equivalent. Moreover, we give some Tauberian theorems for statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -summability.

Keywords

• Statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -limit superior
• Statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -limit inferior
• Statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -bounded
• Statistical A I $\mathfrak{A}^{\mathfrak{I}}$ -Cauchy summability
• Statistical A I ∗ $\mathfrak{A}^{\mathfrak{I}^{\ast }}$ -Cauchy summability
• Tauberian theorem