Difference inequalities and barycentric identities for classical discrete iterated weights

Rutka, Przemysław; Smarzewski, Ryszard

doi:10.1090/mcom/3396

Difference inequalities and barycentric identities for classical discrete iterated weights
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Math. Comp. 88 (2019), 1791-1804 Request permission

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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