Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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A Note on Spliced Sequences and A-density of Points with respect to a Non-negative Matrix

  • Received : 2017.10.12
  • Accepted : 2018.10.10
  • Published : 2019.03.23
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Abstract

For $y{\in}{\mathbb{R}}$, a sequence $x=(x_n){\in}{\ell}^{\infty}$, and a non-negative regular matrix A, Bartoszewicz et. al., in 2015, defined the notion of the A-density ${\delta}_A(y)$ of the indices of those $x_n$ that are close to y. Their main result states that if the set of limit points of ($x_n$) is countable and density ${\delta}_A(y)$ exists for any $y{\in}\mathbb{R}$ where A is a non-negative regular matrix, then ${\lim}_{n{\rightarrow}{\infty}}(Ax)_n={\sum}_{y{\in}{\mathbb{R}}}{\delta}_A(y){\cdot}y$. In this note we first show that the result can be extended to a more general class of matrices and then consider a conjecture which naturally arises from our investigations.

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Acknowledgement

Supported by : UGC, CSIR

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