Approximation results by statistical convergence based on a power series in modular spaces (Q6544240)

Weierstrass Theorem expresses that the set of algebraic polynomials is dense in the space of continuous functions and Korovkin extended Weierstrass's ideas by formulating a generalization known as Korovkin Theorem. Korovkin's work further depends on understanding of approximation theory on the uniform convergence of positive linear operators in \(C[a, b]\), the space of continuous real valued functions defined on \([a, b]\), by testing the convergence only for \(\{1, x, x^{2}\}\) [\textit{W. Kratz} and \textit{U. Stadtmüller}, J. Math. Anal. Appl. 139, No. 2, 362--371 (1989; Zbl 0694.40010)]. This theorem is a fundamental result in the theory of approximation which aims to find simpler functions that can closely approximate more complicated ones. In this paper under review, the authors extend some earlier results in modular spaces via P-statistical convergence [\textit{T. Yurdakadim}, Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 65, No. 2, 65--76 (2016; Zbl 1370.41048)], giving an example to show that Theorem 1 and Theorem 2 are stronger than Theorem 3.2 and Theorem 3.3 in [\textit{C. Bardaro} et al., Appl. Anal. 92, No. 11, 2404--2423 (2013; Zbl 1286.41003)] respectively. The effect of \(P\)-statistical convergence is already shown by providing such examples which may converge in \(P\)-statistical sense but may not be statistically converge and vice versa [\textit{T. Yurdakadim}, Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 65, No. 2, 65--76 (2016; Zbl 1370.41048)].