Restricting statistical convergence
The ``ordinary'' concept of lower- and upper-asymptotic density for a (double) sequence is well known, along with the concept of a sequence being statistically convergent. The authors of the paper under review use an extension of the concept in the following direction: Definition 1. Let \(A\subset\mathbb N\) and denote for every pair \(m,n\in\mathbb N\) the cardinality of the set \(A\cap [m,n]\) by \(A(m,n)\). Let \(0<\alpha\leq 1\) be a real number. Then \[ \underline{\hbox{d}}^{\alpha}(A)= \liminf_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}},\quad \overline{\hbox{d}}^{\alpha}(A)=\limsup_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}}, \] are called lower resp. upper asymptotic density of order \(\alpha\) of the set \(A\). If \(\lim_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}}\) exists, then \(d^{\alpha}(A)=\underline{d}^{\alpha}(A)= \overline{d}^{\alpha}(A)\) is said to be the asymptotic density of order \(\alpha\) of the set \(A\). Definition 2. Let \(x=\{x_k\}_{k\in\mathbb N}\in \omega\) (the space of all real sequences) and \(0<\alpha\leq 1\) be given. Then \(x\) is said to be statistically convergent of order \(\alpha\) if there is a real number \(L\) such that \[ \lim_{n\rightarrow\infty}\,{| \{k\leq n\,:\,| x_k-L| \geq\varepsilon\}| \over n^{\alpha}}=0 \] for all \(\varepsilon>0\). We write \(S^{\alpha}\hbox{--}\lim\,x_k=L\), and the set of all sequences that are statistically convergent of order \(\alpha\) will be denoted by \(m_0^{\alpha}\). The authors then prove: A. If \(\{x_k\}\) is statistically convergent of order \(\alpha\) to \(L\;(0<\alpha<1)\), then there exists a set \(K=\{k_n\,:\, k_1<k_2<\cdots<k_n< \cdots\}\subset\mathbb N\), such that \(\overline{\hbox{d}}^{\alpha}(K)=\infty\) and \(\lim_{n\rightarrow \infty}\,x_{k_n}=L\). B. Let \(0<\alpha\leq\beta\leq 1\), then \(m_0^{\alpha}\subseteq m_0^{\beta}\) and the inclusion is strict for at least those \(\alpha,\beta\) for which there exists a \(k\in\mathbb N\) with \(\alpha<{1\over k}<\beta\). C. Let \(m\) be the set of all bounded real sequences equiped with the \(\sup\)-norm, then for a fixed \(\alpha,\;0<\alpha\leq 1\), the set \(m_0^{\alpha}\cap m\) is a closed linear subspace of \(m\). The second part of the paper studies double sequences, using convergence in Pringsheim's sense; the analogues of the Definitions 1 and 2 and Theorems A, B, C are then given/proved for double sequences \(\{x_{i,j}\}\) which are called convergent to \(\xi\in\mathbb R\) if there exists for any \(\varepsilon>0\) a \(N_{\varepsilon}\in\mathbb N\) with \(| x_{i,j}-\xi| <\varepsilon\) whenever \(i,j\geq N_{\varepsilon}\). The upper/lower densities and other definitions then use in the denominator \(n_1^{\alpha}n_2^{\beta}\) where the two real exponents satisfy \(0<\alpha,\beta\leq 1\). The generalization of Theorem A then uses \(0<\alpha<\beta<1\), that of B uses two pairs \(0<\alpha_1\leq\alpha_2 \leq 1,\,0<\beta_1\leq\beta_2\leq 1\) leading to strict inclusion \(m_0^2(\alpha_1,\beta_1)\subset m_0^2(\alpha_2,\beta_2)\) when there exist \(k,l\in\mathbb N\) with \(\alpha_1<{1\over k}<\alpha_2,\,\beta_1<{1\over l}<\beta_2\). The new Theorem C, finally, states that \(m_0^2(\alpha,\beta)\cap m^2\) is a closed linear subspace of \(m^2\) for \(0<\alpha,\beta\leq 1\). An elegantly written paper with nice results.
statistical convergence of order \(\alpha\)
double density of order \((\alpha,\beta)\)
statistical convergence of order \((\alpha,\beta)\)
scientific article; zbMATH DE number 6006174
scientific article