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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Applications of new Geronimus type identities for real orthogonal polynomials


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 138 (2010), 2125-2134
MSC (2010): Primary 42C05; Secondary 41A17, 41A10, 41A55
Published electronically: February 3, 2010
MathSciNet review: 2596051
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Abstract: Let $ \mu $ be a positive measure on the real line, with associated orthogonal polynomials $ \left\{ p_{n}\right\} $. Let Im$ a\neq 0$. Then there is an explicit constant $ c_{n}$ such that for all polynomials $ P$ of degree at most $ 2n-2$,

$\displaystyle c_{n}\int_{-\infty }^{\infty }\frac{P\left( t\right) }{\left\vert... ...}\left( a\right) p_{n}\left( t\right) \right\vert ^{2}}dt=\int P\text{ }d\mu . $

In this paper, we provide a self-contained proof of this identity. Moreover, we apply the formula to deduce a weak convergence result and a discrepancy estimate, and also to establish a Gauss quadrature associated with $ \mu $ with nodes at the zeros of $ p_{n}\left( a\right) p_{n-1}\left( t\right) -p_{n-1}\left( a\right) p_{n}\left( t\right) $.


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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: http://dx.doi.org/10.1090/S0002-9939-10-10276-7
PII: S 0002-9939(10)10276-7
Keywords: Orthogonal polynomials on the real line, Geronimus formula, discrepancy, weak convergence, Gauss quadrature
Received by editor(s): August 17, 2009
Received by editor(s) in revised form: October 22, 2009
Published electronically: February 3, 2010
Additional Notes: This research was supported by NSF grant DMS0700427 and U.S.-Israel BSF grant 2004353
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society