Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method

Axioms

Tomas Visnyai¹  Robert Vrabel²  Renata Masarova² 

[1] Faculty of Chemical and Food Technology in Bratislava, Slovak University of Technology, Radlinského 9, 812 37 Bratislava, Slovakia;Faculty of Materials Science and Technology in Trnava, Slovak University of Technology, Ulica Jána Bottu č. 2781/25, 917 01 Trnava, Slovakia;

关键词: statistical convergence;    quasi-statistical convergence;    asymptotic density;    quasi-density;    the matrix summability method;   

DOI  :  10.3390/axioms11030088

来源: DOAJ

【 摘 要 】

In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density $d_p\left(A\right)$ of the set $A\subseteq N$ is dependent on the sequence $p=\left(p_n\right)$. Different sequences $\left(p_n\right)$, for the same set $A$, will yield new and distinct densities. If the sequence $\left(p_n\right)$ does not differ from the sequence $\left(n\right)$ in its order of magnitude, i.e., $\underset{n\to \infty }{\lim }\frac{p_n}{n}=1$, then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients.

【 授权许可】

Unknown