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On Spliced Sequences and the Density of Points with Respect to a Matrix Constructed by using a Weight Function

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Ukrainian Mathematical Journal Aims and scope

Following the line of investigation in [Linear Algebra Appl., 487, 22–42 (2015)], for y ∈ ℝ and a sequence x = (xn) ∈ ℓ∞, we define a new notion of density δg with respect to a weight function g of indices of the elements xn close to y, where g : ℕ → [0, ∞)is such that g(n) → ∞and n/g(n)0. We present the relationships between the densities δg of indices of (xn) and the variation of the Cesàrolimit of (xn). Our main result states that if the set of the limit points of (xn) is countable and δg(y) exists for any y ∈ ℝ, then

$$ \underset{n\to \infty }{\lim}\frac{1}{g(n)}\sum \limits_{i-1}^n{x}_i=\sum \limits_{y\in \mathbb{R}}{\delta}_g(y)\cdot y, $$

which is an extended and much more general form of the “natural version of density from the Osikiewicz theorem.” Note that, in [Linear Algebra Appl., 487, 22–42 (2015)], the regularity of the matrix was used in the entire investigation, whereas in the present paper, the investigation is actually performed with respect to a special type of matrix, which is not necessarily regular.

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Authors and Affiliations

  1. Jadavpur University, Kolkata, West Bengal, India

    K. Bose, P. Das & S. Sengupta

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  1. K. Bose
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  2. P. Das
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  3. S. Sengupta
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Correspondence to P. Das.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 9, pp. 1192–1207, September, 2019.

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Bose, K., Das, P. & Sengupta, S. On Spliced Sequences and the Density of Points with Respect to a Matrix Constructed by using a Weight Function. Ukr Math J 71, 1359–1378 (2020). https://doi.org/10.1007/s11253-020-01720-1

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  • DOI: https://doi.org/10.1007/s11253-020-01720-1

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