Abstract
In this paper, we present an overview about algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the coefficients of the three-term recurrence relation they satisfy, the so-called co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. In the case of orthogonal polynomials on the unit circle, we analyze the effects of finite modifications of Verblunsky coefficients on Szegő recurrences. More precisely, we study the structural relations and the corresponding \(\mathcal{C}\)-functions of the orthogonal polynomials with respect to these modifications from the initial ones. By using the Szegő’s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle as well as the relation between the corresponding \(\mathcal{S}\)-functions and \(\mathcal{C}\)-functions is studied.
Keywords
- Orthogonal polynomials on the real line
- Orthogonal polynomials on the unit circle
- Zeros
- Spectral transformations
- Co-polynomials
- Szegő transformation
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References
V.M. Badkov, Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials. Math. Notes 42, 858–863 (1987)
P. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities (Springer, New York, 1995)
K. Castillo, On perturbed Szegő recurrences. J. Math. Anal. Appl. 411, 742–752 (2014)
K. Castillo, Monotonicity of zeros for a class of polynomials including hypergeometric polynomials. Appl. Math. Comput. 266, 173–193 (2015)
K. Castillo, F. Marcellán, J. Rivero, On co-polynomials on the real line. J. Math. Anal. Appl. 427, 469–483 (2015)
K. Castillo, F. Marcellán, J. Rivero, On perturbed orthogonal polynomials on the real line and the unit circle via Szegő’s transformation. Appl. Math. Comput. (2017, Accepted for publication)
T.S. Chihara, On co-recursive orthogonal polynomials. Proc. Am. Math. Soc. 8, 899–905 (1957)
T.S. Chihara, An introduction to orthogonal polynomials, in Mathematics and Its Applications, vol. 13 (Gordon and Breach, New York/London/Paris, 1978)
M.N. de Jesus, J. Petronilho, On orthogonal polynomials obtained via polynomial mappings. J. Approx. Theory 162, 2243–2277 (2010)
J. Dini, P. Maroni, A. Ronveaux, Sur une perturbation de la récurrence vérifiée par une suite de polynômes orthogonaux. Portugal. Math. 46, 269–282 (1989)
W. Erb, Optimally space localized polynomials with applications in signal processing. J. Fourier Anal. Appl. 18 (1), 45–66 (2012)
W. Erb, Accelerated Landweber methods based on co-dilated orthogonal polynomials. Numer. Algorithms 68, 229–260 (2015)
Y.L. Geronimus, On some difference equations and corresponding systems of orthogonal polynomials. Izv. Akad. Nauk SSSR, Ser. Mat. 5, 203–210 (1943)
Y.L. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas and Series of Polynomials Orthogonal on the Unit Circle and on an Interval (Consultants Bureau, New York, 1961)
Y.L. Geronimus, Orthogonal polynomials on a circle and their applications. Am. Math. Soc. Translat. Ser. 1 3, 1–78 (1962)
L. Golinskii, P. Nevai, Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle. Commun. Math. Phys. 223, 223–259 (2001)
M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia in Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)
M. Ismail, X. Li, On sieved orthogonal polynomials IX: orthogonality on the unit circle. Pac. J. Math. 152, 289–297 (1992)
C.G.J. Jacobi, Über die reduction der quadrastischen formen auf die kleinste anzahl glieder. J. Reine Angew. Math. 39, 290–292 (1848)
J. Letessier, Some results on co-recursive associated Laguerre and Jacobi polynomials. SIAM J. Math. Anal. 25 (2), 528–548 (1994)
G.G. Lorentz, M.V. Gollitschek, Y. Makovoz, Constructive Approximation (Springer, New York, 1996)
F. Marcellán, G. Sansigre, Orthogonal polynomials on the unit circle: symmetrization and quadratic decomposition. J. Approx. Theory 65, 109–119 (1991)
F. Marcellán, J.S. Dehesa, A. Ronveaux, On orthogonal polynomials with perturbed recurrence relations. J. Comput. Appl. Math. 30, 203–212 (1990)
L.M. Milne-Thomson, The Calculus of Finite Differences. American Mathematical Society (Chelsea Publishing, Providence, 2000)
G.V. Milovanovic, M.Th. Rassias (eds.), Analytic Number Theory, Approximation Theory and Special Functions (Springer, New York, 2014)
F. Peherstorfer, Finite perturbations of orthogonal polynomials. J. Comput. Appl. Math. 44, 275–302 (1992)
F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials. Constr. Approx. 12, 161–185 (1996)
J. Petronilho, Orthogonal polynomials on the unit circle via a polynomial mapping on the real line. J. Comput. Appl. Math. 216, 98–127 (2008)
A. Ronveaux, Fourth-order differential equations for numerator polynomials. J. Phys. A Math. Gen. 21 (15), L749 (1988)
A. Ronveaux, F. Marcellán, Co-recursive orthogonal polynomials and fourth-order differential equation. J. Comput. Appl. Math. 25 (1), 105–109 (1989)
A. Ronveaux, S. Belmehdi, J. Dini, P. Maroni, Fourth-order differential equation for the co-modified of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 29 (2), 225–231 (1990)
B. Simon, Orthogonal polynomials on the unit circle, Part 1: Classical theory. Colloquium Publications Series, vol. 54 (American Mathematical Society, Providence, 2005)
B. Simon, Orthogonal polynomials on the unit circle, Part 2: Spectral theory. Colloquium Publications Series, vol. 54 (American Mathematical Society, Providence, 2005)
B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials (Princeton University Press, Princeton, 2011)
H.A. Slim, On co-recursive orthogonal polynomials and their application to potential scattering. J. Math. Anal. Appl. 136, 1–19 (1988)
G. Szegő, Orthogonal Polynomials, 4th edn. Colloquium Publications Series, vol. 23 (American Mathematical Society, Providence, 1975)
A. Zhedanov, Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67–86 (1997)
Acknowledgements
The authors wish to express their thanks to Th. M. Rassias and N. J. Daras for the invitation to participate in this volume. The research of the first author is supported by the Portuguese Government through the FCT under the grant SFRH/BPD/101139/2014 and partially supported by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01.
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Castillo, K., Marcellán, F., Rivero, J. (2017). On Co-polynomials on the Real Line and the Unit Circle. In: Daras, N., Rassias, T. (eds) Operations Research, Engineering, and Cyber Security. Springer Optimization and Its Applications, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-51500-7_4
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