Majorization results for zeros of orthogonal polynomials
HTML articles powered by AMS MathViewer
- by Walter Van Assche PDF
- Proc. Amer. Math. Soc. 145 (2017), 3849-3863 Request permission
Abstract:
We show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of $p_n$ and the associated polynomial $p_{n-1}^{(1)}$ and for the zeros of the polynomial obtained by deleting the $k$th row and column $(1 \leq k \leq n)$ in the corresponding Jacobi matrix.References
- Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
- Walter Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. Oxford Science Publications. MR 2061539
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- S. M. Malamud, An analogue of the Poincaré alternation theorem for normal matrices, and the Gauss-Lucas theorem, Funktsional. Anal. i Prilozhen. 37 (2003), no. 3, 85–88 (Russian); English transl., Funct. Anal. Appl. 37 (2003), no. 3, 232–235. MR 2021139, DOI 10.1023/A:1026044902927
- S. M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–4064. MR 2159699, DOI 10.1090/S0002-9947-04-03649-9
- Albert W. Marshall, Ingram Olkin, and Barry C. Arnold, Inequalities: theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. MR 2759813, DOI 10.1007/978-0-387-68276-1
- Rajesh Pereira, Differentiators and the geometry of polynomials, J. Math. Anal. Appl. 285 (2003), no. 1, 336–348. MR 2000158, DOI 10.1016/S0022-247X(03)00465-7
- G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence, RI, 1939 (fourth edition, 1975).
- Walter Van Assche, Orthogonal polynomials, associated polynomials and functions of the second kind, J. Comput. Appl. Math. 37 (1991), no. 1-3, 237–249. MR 1136928, DOI 10.1016/0377-0427(91)90121-Y
Additional Information
- Walter Van Assche
- Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium
- MR Author ID: 176825
- ORCID: 0000-0003-3446-6936
- Email: walter@wis.kuleuven.be
- Received by editor(s): July 12, 2016
- Published electronically: May 24, 2017
- Additional Notes: This research was supported by KU Leuven research grant OT/12/073 and FWO research project G.0864.16N
- Communicated by: Mourad Ismail
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3849-3863
- MSC (2010): Primary 33C45, 42C05; Secondary 26C10, 30C15, 15B51
- DOI: https://doi.org/10.1090/proc/13560
- MathSciNet review: 3665038
Dedicated: Dedicated to T.J. Stieltjes on the occasion of his 160th birthday