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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.
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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