Convex linear combinations of sequences of monic orthogonal polynomials
HTML articles powered by AMS MathViewer
- by A. Cachafeiro and F. Marcellan
- Proc. Amer. Math. Soc. 126 (1998), 2323-2331
- DOI: https://doi.org/10.1090/S0002-9939-98-04272-5
- PDF | Request permission
Abstract:
For a sequence $\{\Phi _n\}_0^\infty$ of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP $\{Q_n\}_0^\infty$ in order that a convex linear combination $\{R_n\}_0^\infty$ with $R_n=\beta \Phi _n+(1-\beta )Q_n$ be a SMOP with respect to a positive measure supported on the unit circle.References
- M. Alfaro, Ma. P. Alfaro, J. J. Guadalupe, and L. Vigil, Correspondance entre suites de polynômes orthogonaux et fonctions de la boule unité de $H^\infty _0(D)$, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 158–163 (French). MR 838980, DOI 10.1007/BFb0076540
- A. Branquinho, L. B. Golinskii, and F. Marcellán, Rational modifications of Lebesgue measure on the unit circle and an inverse problem, submitted.
- Tamás Erdélyi, Paul Nevai, John Zhang, and Jeffrey S. Geronimo, A simple proof of “Favard’s theorem” on the unit circle, Trends in functional analysis and approximation theory (Acquafredda di Maratea, 1989) Univ. Modena Reggio Emilia, Modena, 1991, pp. 41–46. MR 1136583
- Géza Freud, Orthogonale Polynome, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 33, Birkhäuser Verlag, Basel-Stuttgart, 1969 (German). MR 0481888, DOI 10.1007/978-3-0348-7169-3
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- F. Marcellán, F. Peherstorfer, and R. Steinbauer, Orthogonality properties of linear combinations of orthogonal polynomials, Adv. in Comp. Math. 5 (1996), 281–295.
- F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), no. 2, 161–185. MR 1393285, DOI 10.1007/s003659900008
- Franz Peherstorfer and Robert Steinbauer, Perturbation of orthogonal polynomials on the unit circle—a survey, Orthogonal polynomials on the unit circle: theory and applications (Madrid, 1994) Univ. Carlos III Madrid, Leganés, 1994, pp. 97–119. MR 1317108
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Bibliographic Information
- A. Cachafeiro
- Affiliation: Departamento de Matemática Aplicada, E.T.S.I.I., Universidad de Vigo, Spain
- Email: acachafe@dma.uvigo.es
- F. Marcellan
- Affiliation: Departamento de Matemáticas, E.P.S., Universidad Carlos III de Madrid, Spain
- Email: pacomarc@ing.uc3m.es
- Received by editor(s): March 4, 1996
- Received by editor(s) in revised form: January 13, 1997
- Additional Notes: The work of the first author was supported by the DGICYT under grant number PB93-1169.
The work of the second author was supported by an Acción Integrada Hispano-Austriaca 4B/1995. - Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2323-2331
- MSC (1991): Primary 42C05
- DOI: https://doi.org/10.1090/S0002-9939-98-04272-5
- MathSciNet review: 1443374