Triangular \(A\)-statistical approximation by double sequences of positive linear operators (Q889130)

The authors study a variation of \(A\)-statistical convergence of double sequences, called triangular \(A\)-statistical approximation, where, for a suitable matrix \(A=(a_{i,j})\), a double sequence \((x_{i,j})\) is triangular \(A\)-statistically convergent to \(L\) if, for every \(\varepsilon>0\), \[ \lim_{i \to \infty} \sum_{j \in K_i(\varepsilon)}a_{i,j}= 0, \] where \(K_i(\varepsilon) = \{ j\in \mathbb N : j \leq i,\, |x_{i,j}-L|\geq \varepsilon\}\). Examples are provided that illustrate that neither \(A\)-statistical convergence nor triangular \(A\)-statistical convergence imply the other. The authors also prove a Korovkin-type theorem on triangular \(A\)-statistical convergence. They compare this theorem to Korovkin-type theorems related to Pringsheim convergence and \(A\)-statistical convergence of double sequences, giving examples where each theorem can be applied, while the other two theorems can not be applied. The rates of triangular \(A\)-statistical convergence are also studied, and theorems are proven giving approximation properties for summability matrices in terms of these rates of convergence.