Approximation via statistical measurable convergence with respect to power series for double sequences (Q6177648)

In this study, the authors recall the notions of convergence in the Pringsheim sense, statistical convergence for double sequences, convergence for double sequences of real-valued functions, and statistical convergence for double sequences of real-valued functions in the first section. In the next section they introduce the concepts of PMC (product measurable convergent), PMC\(_p\) (product measurable convergent with respect to the power series method), SPMC (statistical product measurable convergent) and SPMC\(_p\) (statistical product measurable convergent with respect to the power series method). They prove with examples that PMC\(_p\) and PMC, SPMC\(_p\) and SPMC, PMC\(_p\) and convergence, and SPMC\(_p\) and statistical convergence cannot be compared, respectively. After that the relationship between PMC\(_p\) and SPMC\(_p\) is given with Theorem 2.11. In the Korovkin-type approximation theorems section, the authors extend the classical Korovkin-type theorem Theorem 3.1 for PMC\(_p\) and SPMC\(_p\) by proving Theorem 3.2 and Theorem 3.4.