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Triangular \(A\)-statistical approximation by double sequences of positive linear operators. (English) Zbl 1338.40009

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.

MSC:

40A35 Ideal and statistical convergence
41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
40B05 Multiple sequences and series
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