An abstract version of the Korovkin theorem via \(\mathcal{A}\)-summation process (Q313418)

Using \(\mathcal{A}\)-summation processes, the authors establish an analog of an abstract version of the Korovkin theorem for a sequence of positive linear operators mapping \(C(X;\mathbb{R})\) into \(C(X;\mathbb{R}),\) where \(X\) is a compact Hausdorff space with at least two points, equipped with the usual norm \(\| f \| = \sup_{x \in X} | f(x) |\), \(f \in C(X;\mathbb{R})\). If \(\mathcal{A} := \{ \mathcal{A}^{(n)} \} = \{ a_{kj}^{(n)} \}\) is a sequence of infinite matrices with non-negative real entries, a sequence \(\{ L_{j} \}\) of positive linear operators of \(C(X;\mathbb{R})\) into \(C(X;\mathbb{R})\) is called \(\mathcal{A}\)-summation process on \(C(X;\mathbb{R})\) if \(\displaystyle\lim_{k \to \infty} \| \sum_{j} a_{kj}^{(n)} L_{j}f - f \| = 0,\) uniformly in \(n\), where it is assumed that the series \(\sum_{j} a_{kj}^{(n)} L_{j}f\) converges for each \(k\), \(n\) and \(f\).