Sequential convergence in the space of absolutely Riemann integrable functions (Q1340782)

The concept of \(R\)-convergence in the space of usual Riemann integrable functions introduced by \textit{W. Dickmeis}, \textit{H. Mevissen}, \textit{R. J. Nessel} and \textit{E. van Wickeren} [J. Approximation Theory 55, No. 1, 65-85 (1988; Zbl 0653.41011)] is extended by the authors to the space of absolutely Riemann integrable functions by establishing several interesting results. The following theorem proved in the paper is noteworthy. Theorem. Let \(f\in R^1\) and let \(|f|\leq g\), \(g\in G\subset R^1\). Then there exists a sequence \(\{f_n\}_{n\in\mathbb{N}}\) of continuous and absolutely Riemann integrable functions such that \(G\)-\(\lim_{n\to\infty} f_n=f\), where \(R^1:= R^1(\mathbb{R}^m)\) is the collection of all absolutely Riemann integrable functions with the property that for all \(g_1,g_2\in G\) there exists \(g_3\) such that \(g_1+ g_2\leq g_3\).