Quadrature identities for interlacing and orthogonal polynomials
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- by D. S. Lubinsky
- Proc. Amer. Math. Soc. 144 (2016), 4819-4829
- DOI: https://doi.org/10.1090/proc/13099
- Published electronically: April 25, 2016
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Abstract:
Let $S$ be a real polynomial of degree $n$ with real simple zeros $\left \{ x_{j}\right \} _{j=1}^{n}$. Let $R$ be a real polynomial of degree $n-1$ whose zeros interlace those of $S$. We prove the quadrature identity \begin{equation*} \int _{-\infty }^{\infty }\frac {P(t)}{S^{2}\left ( t\right ) }h\left ( \frac {R}{S }\left ( t\right ) \right ) dt=\left ( \int _{-\infty }^{\infty }h\left ( t\right ) dt\right ) \sum _{j=1}^{n}\frac {P\left ( x_{j}\right ) }{\left ( RS^{\prime }\right ) \left ( x_{j}\right ) }, \end{equation*} valid for all polynomials $P$ of degree $\leq 2n-2$ and any $h\in L_{1}\left ( \mathbb {R}\right )$. We deduce identities involving orthogonal polynomials and weak convergence results involving orthogonal polynomials.References
- Alan Beardon, Kathy Driver, and Kerstin Jordaan, Zeros of polynomials embedded in an orthogonal sequence, Numer. Algorithms 57 (2011), no. 3, 399–403. MR 2807130, DOI 10.1007/s11075-010-9435-4
- Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, and Olav Njåstad, Orthogonal rational functions, Cambridge Monographs on Applied and Computational Mathematics, vol. 5, Cambridge University Press, Cambridge, 1999. MR 1676258, DOI 10.1017/CBO9780511530050
- René Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types, J. Funct. Anal. 51 (1983), no. 2, 229–258. MR 701057, DOI 10.1016/0022-1236(83)90027-7
- Patrick Dewilde and Harry Dym, Lossless inverse scattering, digital filters, and estimation theory, IEEE Trans. Inform. Theory 30 (1984), no. 4, 644–662. MR 755793, DOI 10.1109/TIT.1984.1056925
- G. Freud, Orthogonal Polynomials, Pergamon Press/Akademiai Kiado, Budapest, 1971.
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
- Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
- Denis Krutikov and Christian Remling, Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure, Comm. Math. Phys. 223 (2001), no. 3, 509–532. MR 1866165, DOI 10.1007/s002200100552
- D. S. Lubinsky, Explicit orthogonal polynomials for reciprocal polynomial weights on $(-\infty ,\infty )$, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2317–2327. MR 2495265, DOI 10.1090/S0002-9939-08-09754-2
- D. S. Lubinsky, Applications of new Geronimus type identities for real orthogonal polynomials, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2125–2134. MR 2596051, DOI 10.1090/S0002-9939-10-10276-7
- D. S. Lubinsky, Old and new Geronimus type identities for real orthogonal polynomials, Jaen J. Approx. 2 (2010), no. 2, 289–301. MR 3222155
- D. S. Lubinsky, New integral identities for orthogonal polynomials on the real line, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1743–1750. MR 2763762, DOI 10.1090/S0002-9939-2010-10601-9
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- Barry Simon, Orthogonal polynomials with exponentially decaying recursion coefficients, Probability and mathematical physics, CRM Proc. Lecture Notes, vol. 42, Amer. Math. Soc., Providence, RI, 2007, pp. 453–463. MR 2352283, DOI 10.1090/crmp/042/23
- Burton Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554–555. MR 131120, DOI 10.1090/S0002-9939-1961-0131120-2
Bibliographic Information
- D. S. Lubinsky
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 116460
- ORCID: 0000-0002-0473-4242
- Email: lubinsky@math.gatech.edu
- Received by editor(s): August 18, 2015
- Received by editor(s) in revised form: January 14, 2016
- Published electronically: April 25, 2016
- Additional Notes: This research was supported by NSF grant DMS1362208
- Communicated by: Walter Van Assche
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4819-4829
- MSC (2010): Primary 41A55, 42C99; Secondary 65D30, 30E05
- DOI: https://doi.org/10.1090/proc/13099
- MathSciNet review: 3544532