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Difference inequalities and barycentric identities for classical discrete iterated weights


Authors: Przemysław Rutka and Ryszard Smarzewski
Journal: Math. Comp. 88 (2019), 1791-1804
MSC (2010): Primary 41A17, 65D05; Secondary 65D32, 41A44, 33D45
DOI: https://doi.org/10.1090/mcom/3396
Published electronically: November 27, 2018
MathSciNet review: 3925485
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Abstract: In this paper we characterize extremal polynomials and the best constants for the Szegő-Markov-Bernstein-type inequalities, associated with iterated weight functions $ \rho _{k}\left ( x\right ) \!=\! A\left ( x+h\right ) \rho _{k-1}\left ( x+h\right ) $ of any classical weight $ \rho _{0}\left ( x\right ) = \rho \left ( x\right ) $ of discrete variable $ x = a+ih$, which is defined to be the solution of a difference boundary value problem of the Pearson type. It yields the effective way to compute numerical values of the best constants for all six basic discrete classical weights of the Charlier, Meixner, Kravchuk, Hahn I, Hahn II, and Chebyshev kind. In addition, it enables us to establish the generic identities between the Lagrange barycentric coefficients and Christoffel numbers of Gauss quadratures for these classical discrete weight functions, which extends to the discrete case the recent results due to Wang et al. and the authors, published in [Math. Comp. 81 (2012) and 83 (2014), pp. 861-877 and 2893-2914, respectively] and [Math. Comp. 86 (2017), pp. 2409-2427].


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Additional Information

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

Ryszard Smarzewski
Affiliation: Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland
Address at time of publication: PWSZ Włocławek, ul. 3 Maja 17, 87-800 Włocławek, Poland
Email: ryszard.smarzewski@gmail.com, rsmax@kul.pl

DOI: https://doi.org/10.1090/mcom/3396
Keywords: Difference Pearson equations, classical iterated weights, discrete orthogonal polynomials, Szeg\H{o}--Markov--Bernstein inequalities, barycentric weights, Christoffel numbers.
Received by editor(s): January 30, 2018
Received by editor(s) in revised form: June 29, 2018
Published electronically: November 27, 2018
Article copyright: © Copyright 2018 American Mathematical Society