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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sobolev orthogonal polynomials: Balance and asymptotics


Authors: Manuel Alfaro, Juan José Moreno-Balcázar, Ana Peña and M. Luisa Rezola
Journal: Trans. Amer. Math. Soc. 361 (2009), 547-560
MSC (2000): Primary 42C05
Published electronically: July 24, 2008
MathSciNet review: 2439416
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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

$\displaystyle \langle P, P \rangle_{\lambda_n}=\int P^2 \, d\mu_0 + \lambda_n \int P'^2 \, d\mu_1, \quad \lambda_n >0, $

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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04536-4
PII: S 0002-9947(08)04536-4
Keywords: Asymptotics, varying Sobolev inner products, potential theory, Mhaskar--Rakhmanov--Saff numbers, Freud weights
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)
Article copyright: © Copyright 2008 American Mathematical Society