On a.e. convergence of Durrmeyer-Stieltjes polynomials
Let \(\nu\) be a finite Borel measure on \([0,1]\). The Durrmeyer-Stieltjes operators are defined by \[ D_ n\nu= (n+1) \sum^ n_{k=0}\left( \int^ 1_ 0 N_{k,n}(t)d\nu\right)N_{k,n}\quad (n\in\mathbb{N}), \] where \(N_{k,n}\) stands for the \(k\)th Bernstein polynomial of degree \(n\). The main results of this paper can be summarized as follows: 1. The maximal operator of the sequence \(\{D_ n\}\) is of weak type. 2. The sequence \(\{D_ n\nu\}\) converges a.e. on \([0,1]\) to the absolutely continuous part of the measure \(\nu\).
scientific article; zbMATH DE number 645056