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

 

Inequalities of Chernoff type for finite and infinite sequences of classical orthogonal polynomials


Authors: Ryszard Smarzewski and Przemysław Rutka
Journal: Proc. Amer. Math. Soc. 138 (2010), 1305-1315
MSC (2000): Primary 26D10; Secondary 33C45, 60E15
Published electronically: November 25, 2009
MathSciNet review: 2578524
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Abstract: In this paper we present two-sided estimates of Chernoff type for the weighted $ L_{w}^{2}$-distance of a smooth function to the $ k$-dimensional space of all polynomials of degree less than $ k$, whenever the weight function $ w$ solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.


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Additional Information

Ryszard Smarzewski
Affiliation: Department of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynów 1h, 20-708 Lublin, Poland
Email: rsmax@kul.lublin.pl

Przemysław Rutka
Affiliation: Department of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynów 1h, 20-708 Lublin, Poland
Email: rootus@kul.lublin.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10150-8
PII: S 0002-9939(09)10150-8
Keywords: Chernoff-type inequalities, classical finite and infinite orthogonal polynomials, optimal constants, generic differential equations.
Received by editor(s): October 27, 2008
Received by editor(s) in revised form: June 17, 2009
Published electronically: November 25, 2009
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.