On asymptotic properties of Freud-Sobolev orthogonal polynomials

  • Authors:

  • Alicia Cachafeiro;Francisco Marcellán;Juan J. Moreno-Balcázar

  • Affiliations:

  • Departamento de Matemática Aplicada, Universidad de Vigo, Spain;Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain;Departamento de Estadística y Matemática Aplicada, Universidad de Almería, La Cañada de San Urbano s/n, 04120 Almeria, Spain and Instituto Carlos I de Física Teórica ...

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2003

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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-x4 dx 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 orthogonal 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(4√nx)} on compact subsets of C\[-a, a] being a = 4√4/3.