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Quadrature identities for interlacing and orthogonal polynomials


Author: D. S. Lubinsky
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
Published electronically: April 25, 2016
MathSciNet review: 3544532
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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

$\displaystyle \int _{-\infty }^{\infty }\frac {P(t)}{S^{2}\left ( t\right ) }h\... ...{P\left ( x_{j}\right ) }{\left ( RS^{\prime }\right ) \left ( x_{j}\right ) },$    

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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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: lubinsky@math.gatech.edu

DOI: https://doi.org/10.1090/proc/13099
Keywords: Orthogonal polynomials on the real line, Geronimus type formula, quadrature formula, interlacing polynomials
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
Article copyright: © Copyright 2016 American Mathematical Society