\(B\)-statistically \(A\)-summability (Q2861315)

\textit{A. R. Freedman} and \textit{J. J. Sember} [Pac. J. Math. 95, 293--305 (1981; Zbl 0504.40002)] generalized the natural density by replacing \(C_{1}\) with an arbitrary nonnegative regular matrix \(A\). A subset \(K\) of \(\mathbb{N}\), the set of positive integers, has \(A\)-density \(\delta_{A}(K)\) if \(\delta_{A}(K) = \lim_{n} \sum_{k\in{K}}{a_{nk}}\) exists.NEWLINENEWLINE\textit{J. Connor} [Can. Math. Bull. 32, No. 2, 194--198 (1989; Zbl 0693.40007)], \textit{J. Connor} and \textit{J. Kline} [J. Math. Anal. Appl. 197, No. 2, 392--399 (1996; Zbl 0867.40001)] and \textit{E. Kolk} [Analysis 13, No. 1--2, 77--83 (1993; Zbl 0801.40005)] extended the idea of statistical convergence to \(A\)-statistical convergence by using the notion of \(A\)-density. A sequence \(\mathbf{x}=(x_{k})\) is said to be \(A\)-statistically convergent to \(L\) if \(\delta_{A}(K(\varepsilon))=0\) for every \(\varepsilon> 0\), where \(K(\varepsilon)=\{k\in{\mathbb{N}}: |x_{k}-L|\geq \varepsilon\}\). The idea of statistical \((C, 1)\)-summability was introduced by \textit{F. Móricz} [J. Math. Anal. Appl. 275, No. 1, 277--287 (2002; Zbl 1021.40002)].NEWLINENEWLINEIn [Math. Comput. Modelling 49, No. 3--4, 672--680 (2009; Zbl 1182.40004)], \textit{O. H. H. Edely} and \textit{M. Mursaleen} generalized these statistical summability methods by defining the notion of statistical \(A\)-summability and studied its relationship with \(A\)-statistical convergence. A sequence \(\mathbf{x}\) of points in \(\mathbb{R}\) is called statistically \(A\)-summable to \(L\) if, for every \(\varepsilon>0\), \(\delta_{A}(\{ i \leq n: |y_{i}-L|\geq {\varepsilon} \})=0\), where \(y_{i}=A_{i}(\mathbf{x})\). Thus \(\mathbf{x}=(x_{k})\) is statistically \(A\)-summable to \(L\) if and only if \(A(\mathbf{x})\) is statistically convergent to \(L\). Let \(A=(a_{ij})\) and \(B=(b_{nk})\) be two nonnegative regular matrices. A sequence \(\mathbf{x} = (x_{k})\) of real numbers is said to be \(B\)-statistically \(A\)-summable to \(L\) if, for every \(\varepsilon> 0\), the set \(K_{\varepsilon}=\{i: |y_{i}-L|\geq\varepsilon \}\) has \(B\)-density zero.NEWLINENEWLINEThe author establishes some relations between \(B\)-summability, \(B\)-statistical \(A\)-summability and statistical \(A\)-summability. He also introduces and investigates \(B\)-statistical \(A\)-cluster points of a sequence.