On asymptotic properties of Freud-Sobolev orthogonal polynomials

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Abstract

In this paper we consider a Sobolev inner product (f, g)s= ∫ fg dμ + λ ∫ f′ g′ dμ 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 dμ = e-x4dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight e-x4) and the Sobolev orthongal polynomials Qn. Finally, we obtain some asymptotics for {Qn}. More precisely, we give the relative asymptotics {Qn(x)/Pn(x)} on compact subsets of ℂ\ℝ as well as the outer Plancherel-Rotach-type asymptotics {Qn(4 nx)/Pn(4nx)} on compact subsets of ℂ\[-a,a] being a = 4 4/3. © 2003 Elsevier Inc. All rights reserved.

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Cachafeiro, A., Marcellán, F., & Moreno-Balcázar, J. J. (2003). On asymptotic properties of Freud-Sobolev orthogonal polynomials. Journal of Approximation Theory, 125(1), 26–41. https://doi.org/10.1016/j.jat.2003.09.003

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