## 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 $\lambda _n .$## References

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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