Sobolev orthogonal polynomials: Balance and asymptotics
HTML articles powered by AMS MathViewer
- 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
- M. Alfaro, A. Martínez-Finkelshtein, and M. L. Rezola, Asymptotic properties of balanced extremal Sobolev polynomials: coherent case, J. Approx. Theory 100 (1999), no. 1, 44–59. MR 1710552, DOI 10.1006/jath.1998.3336
- Manuel Alfaro, Juan J. Moreno-Balcázar, Teresa E. Pérez, Miguel A. Piñar, and M. Luisa Rezola, Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 141–150. MR 1858274, DOI 10.1016/S0377-0427(00)00639-7
- Alicia Cachafeiro, Francisco Marcellán, and Juan J. Moreno-Balcázar, On asymptotic properties of Freud-Sobolev orthogonal polynomials, J. Approx. Theory 125 (2003), no. 1, 26–41. MR 2016838, DOI 10.1016/j.jat.2003.09.003
- J. S. Geronimo, D. S. Lubinsky, and F. Marcellan, Asymptotics for Sobolev orthogonal polynomials for exponential weights, Constr. Approx. 22 (2005), no. 3, 309–346. MR 2164139, DOI 10.1007/s00365-004-0578-1
- Eli Levin and Doron S. Lubinsky, Orthogonal polynomials for exponential weights, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 4, Springer-Verlag, New York, 2001. MR 1840714, DOI 10.1007/978-1-4613-0201-8
- G. López and E. A. Rakhmanov, Rational approximations, orthogonal polynomials and equilibrium distributions, Orthogonal polynomials and their applications (Segovia, 1986) Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 125–157. MR 973424, DOI 10.1007/BFb0083356
- Francisco Marcellán and Juan José Moreno Balcázar, Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports, Acta Appl. Math. 94 (2006), no. 2, 163–192. MR 2273888, DOI 10.1007/s10440-006-9073-y
- A. Martínez-Finkelshtein, Bernstein-Szegő’s theorem for Sobolev orthogonal polynomials, Constr. Approx. 16 (2000), no. 1, 73–84. MR 1848842, DOI 10.1007/s003659910003
- Andrei Martínez-Finkelshtein, Juan J. Moreno-Balcázar, Teresa E. Pérez, and Miguel A. Piñar, Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures, J. Approx. Theory 92 (1998), no. 2, 280–293. MR 1604939, DOI 10.1006/jath.1997.3123
- L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan & Co., Ltd., London, 1951. MR 0043339
- Juan José Moreno-Balcázar, Smallest zeros of some types of orthogonal polynomials: asymptotics, J. Comput. Appl. Math. 179 (2005), no. 1-2, 289–301. MR 2134372, DOI 10.1016/j.cam.2004.09.045
- Paul Nevai, Orthogonal polynomials associated with $\textrm {exp}(-x^{4})$, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982) CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 263–285. MR 729336, DOI 10.1137/0514048
- W. Rudin, Real and Complex Analysis, McGraw–Hill, New York, 1986.
- Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
- Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828, DOI 10.1017/CBO9780511759420
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