On matrix summability of spliced sequences and \(A\)-density of points
For a non-negative regular matrix \(A\) and \(E\subset {\mathbb{N}}\), the upper and lower \(A\)-densities of \(E\), denoted by \(\overline{\delta_{A}}(E)\) and \(\underline{\delta_{A}}(E)\), are defined as \(\overline{\delta_{A}}(E) = \lim_{x\rightarrow \infty} \sum_{k\in{E}}a_{n,k}\) and \(\underline{\delta_{A}}(E)= \liminf_{n\rightarrow\infty} \sum_{k\in{E}}a_{n,k}\), respectively. If \(\overline{\delta_{A}}(E) = \underline{\delta_{A}}(E)\), then the \(A\)-density of \(E\) exists and it is denoted by \(\delta_{A}(E)\). Using the above idea, the authors introduce the notions of upper \(A^{(x_{n})}\)-density and lower \(A^{(x_{n})}\)-density of a real number \(y\) with respect to a fixed bounded sequence \((x_{n})\) as \[ \overline{\delta_{A^{(x_{n})}}}(y) = \lim _{\varepsilon\rightarrow 0^{+}} \overline{\delta_{A}}(\{n: |x_{n}-y|\leq\varepsilon\}) \] and \[ \underline{\delta_{A^{(x_{n})}}}(y) = \lim _{\varepsilon\rightarrow 0^{-}} \underline{\delta_{A}}(\{n: |x_{n}-y|\leq\varepsilon\}), \] respectively. If \(\overline{\delta_{A^{(x_{n})}}}(y)=\underline{\delta_{A^{(x_{n})}}}(y)\), then the common value is denoted by \(\delta_{A}^{(x_{n})}(y)\); the authors write \(\delta_{A}(y)\) instead of \(\delta_{A}^{(x_{n})}(y)\) since it is always clear which sequence \((x_{n})\) is considered. In their approach, their definition for a sequence \((x_{n})\), a density \(\delta_{A}(y)\) of indices of those \((x_{n})\) which are close to \(y\), which is a new idea not dealt with so far in the literature, is a more general approach than that of \textit{J. A. Osikiewicz} [Rocky Mt. J. Math. 35, No. 3, 977--996 (2005; Zbl 1080.40001)]. Their investigation is not at all analogous to that of Osikiewicz and involves essentially new methods of proofs. They find connections between \(A\)-densities \(\delta_{A}(y)\) of indices of \((x_{n})\) and the \(A\)-limit of \((x_{n})\). Their main result states that, if the set of limit points of a bounded sequence \((x_{n})\) is countable and \(\delta_{A}(y)\) exists for any \(y \in { \mathbb{R} }\), then \(\lim_{n\rightarrow\infty} (Ax)_{n}= \sum^{}_{y\in{\mathbb{R}} }\delta_{A}(y) . y\), which presents a different view of Osikiewicz's theorem [loc. cit.].
scientific article; zbMATH DE number 6506842
scientific article