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Hermite-Fejér type interpolation and Korovkin's theorem

Published online by Cambridge University Press:  17 April 2009

H.-B. Knoop  and
F. Locher
H.-B. Knoop
Affiliation:
Universität Duisburg, Fachbereich Mathematik, Postfach 101629, D-4100 Duisburg, West Germany
F. Locher
Affiliation:
Fern Universität, Hagen Fachbereich Mathematik und Informatik, Postfach 940, D-5800 Hagen, West Germany
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Abstract

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In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 3 , December 1990 , pp. 383 - 390
Copyright
Copyright © Australian Mathematical Society 1990

References

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