## Annals of Functional Analysis

### The $\lambda ^{+r}(\mu )$-statistical convergence

#### Abstract

Let $\lambda =(\lambda _{n})_{n\geq 1}$ be a nondecreasing sequence of positive numbers tending to infinity such that $\lambda _{1}=1$ and $\lambda_{n+1}\leq \lambda _{n}+1$ for all $n$, and let $I_{n}=[n-\lambda _{n}+1,n]$ for $n=1,2,\ldots$. Then for any given nonzero sequence $\mu$, we define by $\Delta ^{+}(\mu)$ the operator that generalizes the operator of the first difference and is defined by $\Delta ^{+}(\mu )x_{k}=\mu_{k}(x_{k}-x_{k+1})$. In this article, for any given integer $r\geq 1$, we deal with the $\lambda ^{+r}(\mu )$ -statistical convergence that generalizes in a certain sense the well-known $\lambda _{E}^{r}$-statistical convergence. The main results consist in determining sets of sequences $\chi$ and $\chi ^{\prime }$ of the form $s_{\xi }^{0}$ satisfying $\chi \subset [V,\lambda ]_{0}(\Delta ^{+r}(\mu ))\subset \chi^{\prime }$ and sets $\kappa$ and $\kappa^{\prime }$ of the form $s_{\xi }$ satisfying $\kappa \leq [V,\lambda ]_{\infty }(\lambda^{+r}(\mu ))\leq \kappa ^{\prime }$. This study is justified since the infinite matrix associated with the operator $\Delta ^{+r}(\mu )$ cannot be explicitly calculated for all ${r}$.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 1 (2017), 1-15.

Dates
Accepted: 14 May 2016
First available in Project Euclid: 14 October 2016

https://projecteuclid.org/euclid.afa/1476450343

Digital Object Identifier
doi:10.1215/20088752-3720471

Mathematical Reviews number (MathSciNet)
MR3558300

Zentralblatt MATH identifier
06667765

#### Citation

de Malafosse, B.; Mursaleen, M.; Rakočević, V. The $\lambda ^{+r}(\mu )$ -statistical convergence. Ann. Funct. Anal. 8 (2017), no. 1, 1--15. doi:10.1215/20088752-3720471. https://projecteuclid.org/euclid.afa/1476450343