A note on some generalized summability methods (Q2850098)

The idea of statistical convergence was formerly given under the name ``almost convergence'' by Antoni Zygmund in the first edition of his celebrated monograph [Trigonometrical series. Warszawa, Lwow: Subwencji Fundusz Kultury Narodowej (1935; Zbl 0011.01703)]. The concept was formally introduced by \textit{H. Fast} [Colloq. Math. 2, 241--244 (1951; Zbl 0044.33605)] and was later reintroduced by \textit{I. J. Schoenberg} [Am. Math. Mon. 66, 361--375, 562--563 (1959; Zbl 0089.04002)], and also independently by \textit{R. C. Buck} [Am. J. Math. 75, 335-346 (1953; Zbl 0050.05901)]. The concept of statistical convergence is a generalization of the usual notion of convergence for real-valued sequences that parallels the usual theory of convergence. A sequence \((x_{k})\) of points in \(\mathbb R\) is called statistically convergent to an element \(L\) of \(\mathbb R\) if for each \(\varepsilon >0\) \(\lim _{n\rightarrow \infty}\frac {1}{n}| \{k\leq n\!: | x_{k}-L| \geq {\varepsilon}\}| =0\). The concept of \(A\)-statistical convergence via a nonnegative regular summability matrix \(A\) was introduced and studied by \textit{E. Kolk} [Proc. Est. Acad. Sci., Phys. Math. 45, No. 2--3, 187--192 (1996; Zbl 0865.40001); corrigendum ibid. 46, No. 1--2, 150 (1997; Zbl 0906.40002)]. The notion of statistical convergence was further extended to \(I\)-convergence by \textit{P. Kostyrko, T. Šalát} and \textit{W. Wilczyński} [Real Anal. Exch. 26(2000--2001), No. 2, 669--685 (2001; Zbl 1021.40001)], and also independently by \textit{F. Nuray} and \textit{W. H. Ruckle} [J. Math. Anal. Appl. 245, No. 2, 513--527 (2000; Zbl 0955.40001)].NEWLINENEWLINEIn the paper under review, the authors introduce the notion of \(A^{I}\)-summability using a nonnegative regular summability matrix \(A\) in the sense that a sequence \(\mathbf {x}=(x_{k})\) of points in \(\mathbb R\) is called \(A^{I}\)-summable to a real number \(L\) if the sequence \(\bigl (A_{n}(\mathbf {x})\bigr)\) is \(I\)-convergent to \(L\), and the notion of \(A^{I}\)-statistical convergence in the sense that \(\mathbf {x}=(x_{k})\) is \(A^{I}\)-statistically convergent to \(L\) if for any \(\varepsilon >0\) and \(\delta >0\) \(\bigl \{n\in {\mathbb {N}}\!: \sum _{k\in {K(\varepsilon)}}a_{nk} \geq {\delta} \bigr \}\in {I}\), where \(K(\varepsilon)=\{k\in {\mathbb {N}}\!: | x_{k}-L| \geq \varepsilon \}\). They prove that the set of all bounded and \(A^{I}\)-statistically convergent sequences is a closed subset of the set of all bounded sequences, any bounded \(A^{I}\)-statistically convergent sequence is \(A^{I}\)-summable to the same limit, and give an ideal version of the dominated convergence theorem. They present interesting examples and counterexamples as well.