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On asymptotic properties of Freud–Sobolev orthogonal polynomials

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2003-11
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Elsevier
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Abstract
In this paper we consider a Sobolev inner product $(f,g)_S=\int fg\,d\mu+ \lambda \int f'g'\,d\mu (*)$, and we characterize the measures μ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure μ and the polynomials, {Qn}, orthogonal with respect to (*), such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a Freud weight. In particular, we study the case $d\mu=e^{-x^4}dx$ supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight $e^{-x^4}$)and the Sobolev orthogonal polynomials Qn. Finally, we obtain some asymptotics for {Qn}.
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16 pages, no figures.-- MSC2000 codes: 33C45; 33C47; 42C05.
MR#: MR2016838 (2005e:33004)
Zbl#: Zbl 1043.33005
Keywords
Sobolev orthogonal polynomials, Freud polynomials, Asymptotics
Bibliographic citation
Journal of Approximation Theory, 2004, vol. 125, n. 1, p. 26-41