Berens, H.; Xu, Y. On Bernstein-Durrmeyer polynomials with Jacobi weights. (English) Zbl 0715.41013 Approximation theory and functional analysis. In honor of George G. Lorentz on the occasion of his 80th birthday, Sev. Pap. Conf., Austin/TX (USA) 1990, 25-46 (1991). Summary: [For the entire collection see Zbl 0715.00016.] The approximation behavior of the Bernstein-Durrmeyer polynomials with respect to the Jacobi weights for the weighted \(L^ p\)-spaces, \(1\leq p\leq \infty\) is studied. The polynomials can be identified with the de la Vallée-Poussin means of the Jacobi series of the associated function. These means have special properties which allow us to give a complete characterization of the approximation behavior by use of the Peetre K- modulus between the Lebesgue spaces and weighted Sobolev spaces for \(1<p<\infty\). For \(p=1\) and \(p=\infty\), we have only partial results. In an additional paragraph we point out the close relationship between the behavior of the Kantorovič and the Bernstein-Durrmeyer polynomials. Cited in 1 ReviewCited in 46 Documents MSC: 41A10 Approximation by polynomials Keywords:Bernstein-Durrmeyer polynomials; Jacobi weights; weighted \(L^ p\)- spaces; de la Vallée-Poussin means; Jacobi series Citations:Zbl 0715.00016 PDFBibTeX XML