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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.

MSC:

41A10 Approximation by polynomials

Citations:

Zbl 0715.00016