Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convex linear combinations of sequences
of monic orthogonal polynomials

Authors: A. Cachafeiro and F. Marcellan
Journal: Proc. Amer. Math. Soc. 126 (1998), 2323-2331
MSC (1991): Primary 42C05
MathSciNet review: 1443374
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. 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^{∞}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,
  • 2. A. Branquinho, L. B. Golinskii, and F. Marcellán, Rational modifications of Lebesgue measure on the unit circle and an inverse problem, submitted.
  • 3. 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
  • 4. Géza Freud, Orthogonale Polynome, Birkhäuser Verlag, Basel-Stuttgart, 1969 (German). Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 33. MR 0481888
  • 5. Ya L. Geronimus, Polynomials Orthogonal on a circle and their applications, Amer. Math. Soc. Transl. (1), vol. 3, Providence, Rhode Island (1962), 1-78. MR 15:869i
  • 6. F. Marcellán, F. Peherstorfer, and R. Steinbauer, Orthogonality properties of linear combinations of orthogonal polynomials, Adv. in Comp. Math. 5 (1996), 281-295. CMP 97:02
  • 7. 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,
  • 8. 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
  • 9. Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C05

Retrieve articles in all journals with MSC (1991): 42C05

Additional Information

A. Cachafeiro
Affiliation: Departamento de Matemática Aplicada, E.T.S.I.I., Universidad de Vigo, Spain

F. Marcellan
Affiliation: Departamento de Matemáticas, E.P.S., Universidad Carlos III de Madrid, Spain

Keywords: Orthogonal polynomials, C-functions, measures on the unit circle
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
Article copyright: © Copyright 1998 American Mathematical Society