Pointwise approximation of integrable functions by the Durrmeyer polynomials (Q1976033)

The author obtains estimates on the degree of approximation at a Lebesgue-Denjoy point, of a Denjoy-Perron integrable function \(f\) on \([0,1]\), by means of the Durrmeyer polynomials \(M_nf\), associated with it. For \(f\in L[0,1]\), a result by \textit{M. M. Derriennic} [J. Approximation Theory 31, 325-343 (1981; Zbl 0475.41025)] about the convergence of \(M_nf(x)\) to \(f(x)\), is now a corollary. If \(f\) is of bounded \(p\)th-variation, then the author obtains estimates involving the local \(p\)th-variation. He also shows that for functions \(f\) such that the local \(p\)th-variation is in Lip \(\alpha\), the estimates are best possible.