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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Sobolev orthogonal polynomials: Balance and asymptotics
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by Manuel Alfaro, Juan José Moreno-Balcázar, Ana Peña and M. Luisa Rezola PDF
Trans. Amer. Math. Soc. 361 (2009), 547-560 Request permission

Abstract:

Let $\mu _0$ and $\mu _1$ be measures supported on an unbounded interval and $S_{n,\lambda _n}$ the extremal varying Sobolev polynomial which minimizes \begin{equation*} \langle P, P \rangle _{\lambda _n}=\int P^2 d\mu _0 + \lambda _n \int P’^2 d\mu _1, \quad \lambda _n >0, \end{equation*} in the class of all monic polynomials of degree $n$. The goal of this paper is twofold. On the one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence $(\lambda _n)$ such that both measures $\mu _0$ and $\mu _1$ play a role in the asymptotics of $\left (S_{n, \lambda _n} \right ).$ On the other hand, we apply such ideas to the case when both $\mu _0$ and $\mu _1$ are Freud weights. Asymptotics for the corresponding $S_{n, \lambda _n}$ are computed, illustrating the accuracy of the choice of $\lambda _n .$
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Additional Information
  • Manuel Alfaro
  • Affiliation: Departamento de Matemáticas and IUMA, Universidad de Zaragoza, c/Pedro Cerbuna 12, 50009 Zaragoza, Spain
  • Juan José Moreno-Balcázar
  • Affiliation: Departamento de Estadística y Matemática Aplicada, Universidad de Almería, La Canada de San Urbano, 04120 Almeria, Spain – and – Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada, Spain
  • Ana Peña
  • Affiliation: Departamento de Matemáticas and IUMA, Universidad de Zaragoza, c/Pedro Cerbuna 12, 50009 Zaragoza, Spain
  • M. Luisa Rezola
  • Affiliation: Departamento de Matemáticas and IUMA, Universidad de Zaragoza, c/Pedro Cerbuna 12, 50009 Zaragoza, Spain
  • Email: rezola@unizar.es
  • Received by editor(s): June 26, 2006
  • Received by editor(s) in revised form: October 19, 2006, and April 26, 2007
  • Published electronically: July 24, 2008
  • Additional Notes: The first author was partially supported by MEC of Spain under Grant MTM2006-13000-C03-03, FEDER funds (EU), and the DGA project E-64 (Spain)
    The second author was partially supported by MEC of Spain under Grant MTM2005–08648–C02–01 and Junta de Andalucía (FQM229 and excellence projects FQM481, PO6-FQM-1735)
    The third author was partially supported by MEC of Spain under Grants MTM 2004-03036 and MTM2006-13000-C03-03, FEDER funds (EU), and the DGA project E-64, Spain
    The fourth author was partially supported by MEC of Spain under Grant MTM2006-13000-C03-03, FEDER funds (EU), and the DGA project E-64 (Spain)
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 547-560
  • MSC (2000): Primary 42C05
  • DOI: https://doi.org/10.1090/S0002-9947-08-04536-4
  • MathSciNet review: 2439416