A Quantitative Bohman–Korovkin Theorem and its Sharpness for Riemann Integrable Functions

Abstract

Inspired by work of Pólya (1933) on the convergence of quadrature formulas, we previously introduced a concept of sequential convergence in the space of Riemann integrable functions under which this classical space is in fact the completion of continuous functions. Discussing its consequences to approximation theory, among others we already extended the (qualitative as well as quantitative) Bohman–Korovkin theorem from continuous to Riemann integrable functions. The error bounds obtained were given in terms of the first $\tau$-modul. One purpose of this paper is to derive corresponding results, involving the second $\tau$-modul. As a consequence of our previous quantitative uniform boundedness principle it is then shown that these estimates are indeed sharp. The general results obtained are illustrated in connection with Bernstein polynomials and linear interpolating splines.