Abstract
We briefly review some old and new results on zeros of orthogonal polynomials for weights on (-∞, ∞). In addition, we discuss the zeros of some special biorthogonal polynomials.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buyarov, V.S.: 1991, ‘Logarithmic Asymptotics of Polynomials Orthogonal on the Real Axis’, Soviet Math. Dokt.Vol. no. 43, pp. 743–746.
Buyarov, V.S.: 1992, ‘Logarithmic Asymptotics of Polynomials Orthogonal on ℝ with Asymmetric Weights’, Notes of the Academy of Sciences of the USSR, Vol. no. 50, pp. 789–795.
Criscuolo, G., Della Vecchia, B., Lubinsky, D.S. and Mastroianni, G.: ‘Functions of the Second Kind for Freud Weights and Series Expansions of Hilbert Transforms’, submitted.
Driver, K.A.: 1991, ‘Nth Root Asymptotics for Extremal Errors Associated with Slowly Decreasing Weights’, Quaestiones Mathematicae,Vol. no. 14, pp. 337–354.
Erdös, P.: 1972, ‘On the Distribution of the Roots of Orthogonal Polynomials’, in: Proceedings of the Conference on the Constructive Theory of Functions (eds. G. Alexits, et. al.), Akademiai Kiado, Budapest, pp. 145–150.
Freud, G.: 1971, ‘Orthogonal Polynomials’, Pergamon Press/Akademiai Kiado, Budapest/Oxford.
Freud, G., Giroux, A. and Rahman, Q.I.: 1978, ‘On Approximation by Polynomials with Weight exp(-|x|)’, Caned. J. Math.Vol. no. 30, pp. 358–372 (in French).
Goncar, A.A. and Rahmanov, E.A.: 1986, ‘Equilibrium Measure and The Distribution of Zeros of Extremal Polynomials’, Math. USSR. Sbornik, Vol. no. 53, pp. 119–130.
Levin, A.L. and Lubinsky, D.S.: 1992, ‘Christoffel Functions, Orthogonal Polynomials, and Nevai’s Conjecture for Freud Weights’, Constructive Approximation, Vol. no. 8, pp. 463–535.
Levin, A.L. and Lubinsky, D.S.: ‘Orthogonal Polynomials and Christoffel Functions for exp(-|x|α), α ≤ 1, J. Approx. Theory, to appear.
Lubinsky, D.S., Mhaskar, H.N. and Saff, E.B.: 1988, ‘A Proof of Freud’s Conjecture for Exponential Weights’, Constructive Approximation, Vol. no. 4, pp. 65–83.
Lubinsky, D.S., and Saff, E.B.: 1988, ‘Strong Asymptotics for Extremal Polynomials Associated with Exponential Weights’, Springer Lecture Notes in Mathematics, Vol. no. 1305, Springer, Berlin.
Lubinsky, D.S. and Sidi, A.: ‘Strong Asymptotics for Polynomials Biorthogonal to Powers of log x’, submitted.
Magnus, Al.: 1993, ‘Refined Asymptotics for Freud Recurrence Coefficients’, in: Collection of Unsolved Problems of Linear and Complex Analysis (eds. V. Havin & N. Nikolskii), Springer Lecture Notes, Springer.
Máté, A., Nevai, P. and Totik, V.: 1986, ‘Asymptotics for the Zeros of Orthogonal Polynomials Associated with hnfinite Intervals’, J. London Math. Soc. Vol. no. 33, pp. 303–310.
Máté, A., Nevai, P. and Zaslaysky, T.: 1985, ‘Asymptotic Expansion of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights’, Trans. Amer. Math. Soc. Vol. no. 287, pp. 495–505.
Mhaskar, H.N.: 1987, ‘Some Discrepancy Theorems’, in: Approximation Theory, Tampa (ed. E.B. Saff), Springer Lecture Notes, Vol. 1287, Springer, New York, pp. 117–131.
Mhaskar, H.N. and Saff, E.B.: 1984, ‘Extremal Problems for Polynomials with Exponential Weights’, Trans. Amer. Math. Soc.Vol. no. 285, pp. 203–234.
Mhaskar, H.N. and Saff, E.B.: 1985, ‘Where Does the Sup-Norm of a Weighted Polynomial Live?’, Constructive Approximation, Vol. no. 1, pp. 71–91.
Mhaskar, H.N. and Saff, E.B.: 1987, ‘Where Does the Lp-Norm of a Weighted Polynomial Live?’, Trans. of the Amer. Math. Soc. Vol. no. 303, pp. 109–124.
Nevai, P.: 1986, ‘Geza Freud, Orthogonal Polynomials and Christoffel Functions: A Case Study’, J. Approx. Theory, Vol. no. 48, pp. 3–167.
Nevai, P. and Dehesa, J.S.: 1979, ‘On Asymptotic Average Properties of Zeros of Orthogonal Polynomials’, SIAM J. Math. Anal. Vol. no. 10, pp. 1184–1192.
Rahmanov, E.A.: 1984, ‘On Asymptotic Properties of Polynomials Orthogonal on the Real Axis’, Math. USSR. Sbornik, Vol. no. 47, pp. 155–193.
Saff, E.B. and Totik, V.: ‘Logarithmic Potential with External Fields’, Springer-Verlag, in press.
Sidi, A. and Levin, D.: 1983, ‘Prediction Properties of the t-Transformation’, SIAM J. Num. Anal. Vol. no. 20, pp. 589–598.
Stahl, H. and Totik, V.: 1992, ‘General Orthogonal Polynomials, Encyclopaedia of Mathematics and its Applications’, Cambridge University Press, Vol. no. 43.
Szegö, G.: 1975, ‘Orthogonal Polynomials’, American Mathematical Society Colloquium Publications, Vol. no. 23, American Mathematical Society, Providence, Rhode Island, Fourth edition.
Ullman, J.L.: 1990, ‘On Orthogonal Polynomials Associated with The Infinite Interval’, in: Approximation Theory III, (ed. E.W. Cheney), Academic Press, New York, pp. 889–895.
Van Assche, W.: 1985, ‘Weighted Zero Distribution for Polynomials Orthogonal on an Infinite Interval’, SIAM J. Math. Anal. Vol. no. 16, pp. 1317–1334.
Van Assche, W.: 1987, ‘The Ratio of q-like Orthogonal Polynomials’, J. Math. Anal. Appins. Vol. no. 128, pp. 535–547.
Van Assche, W.: 1989, ‘Norm Behaviour and Zero Distribution for Orthogonal Polynomials with Non-Symmetric Weights’, Constr. Approx. Vol. no. 5, pp. 329–345.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Lubinsky, D.S. (1994). Zeros of Orthogonal and Biorthogonal Polynomials: Some Old, Some New. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_1
Download citation
DOI: https://doi.org/10.1007/978-94-011-0970-3_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4420-2
Online ISBN: 978-94-011-0970-3
eBook Packages: Springer Book Archive