Lacunary Strong Convergence of Difference Sequences With Respect to a Modulus Function


Rifat Çolak




A sequence $\theta=(k_r)$ of positive integers is called lacunary if $k_0=0$, $0<k_r<k_{r+1 }$ and $h_r=k_r-k_{r-1}\rightarrow\infty$ as $r\rightarrow1$. The intervals determined by $\theta$ are denoted by $I_r=(k_{r-1},k_r]$. Let $\omega$ be the set of all sequences of complex numbers and $f$ be a modulus function. Then we define \begin{align*} N_\theta(\Delta^m,f)&=\{x\in\omega:\lim_r\frac1{h_r}\sum_{k\in I_r}f(|\Delta^mx_k-l|)=0\text{ for some }l\}.\\ N^0_\theta(\Delta^m,f)&=\{x\in\omega:\lim_r\frac1{h_r}\sum_{k\in I_r}f(|\Delta^mx_k|)=0\}.\\ N^\infty_\theta(\Delta^m,f)&=\{x\in\omega:\sup_r\frac1{h_r}\sum_{k\in I_r}f(|\Delta^mx_k|)<\infty\}. \end{align*} where $\Delta x_k=x_k-x_{k+1}$, $\Delta^mx_k=\Delta^{m-1}x_k-\Delta^{m-1}x_{k+1}$ and $m$ is a fixed positive integer. In this study we give various properties and inclusion relations on these sequence spaces.