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
HTML articles powered by AMS MathViewer

by Przemysław Rutka and Ryszard Smarzewski;

Math. Comp. 88 (2019), 1791-1804

DOI: https://doi.org/10.1090/mcom/3396

Published electronically: November 27, 2018

HTML | PDF | 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].

References

Ravi P. Agarwal and Gradimir V. Milovanović, Extremal problems, inequalities, and classical orthogonal polynomials, Appl. Math. Comput. 128 (2002), no. 2-3, 151–166. Orthogonal systems and applications. MR 1891017, DOI 10.1016/S0096-3003(01)00070-4
W. A. Al-Salam and T. S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3 (1972), 65–70. MR 316772, DOI 10.1137/0503007
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884
Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 380189
P. Dörfler, New inequalities of Markov type, SIAM J. Math. Anal. 18 (1987), no. 2, 490–494. MR 876288, DOI 10.1137/0518039
A. Erdélyi, Higher Transcendental Functions, vol. II, McGraw-Hill Book Company, Inc., 1953.
Geza Freud, Markov-Bernstein type inequalities in $L_{p}(-\infty ,\infty )$, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York-London, 1976, pp. 369–377. MR 430187
Géza Freud, On Markov-Bernstein-type inequalities and their applications, J. Approximation Theory 19 (1977), no. 1, 22–37. MR 425426, DOI 10.1016/0021-9045(77)90026-0
A. G. García, F. Marcellán, and L. Salto, A distributional study of discrete classical orthogonal polynomials, Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992), 1995, pp. 147–162. MR 1340932, DOI 10.1016/0377-0427(93)E0241-D
Allal Guessab and Gradimir V. Milovanović, Weighted $L^2$-analogues of Bernstein’s inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994), no. 1, 244–249. MR 1265894, DOI 10.1006/jmaa.1994.1078
Charles Jordan, Calculus of finite differences, 3rd ed., Chelsea Publishing Co., New York, 1965. Introduction by Harry C. Carver. MR 183987
I. H. Jung, K. H. Kwon, and D. W. Lee, Markov type inequalities for difference operators and discrete classical orthogonal polynomials, Advances in difference equations (Veszprém, 1995) Gordon and Breach, Amsterdam, 1997, pp. 335–342. MR 1636338
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
Wolfram Koepf and Dieter Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998), no. 1, 57–94. MR 1627168, DOI 10.1016/S0377-0427(98)00023-5
Wolfram Koepf and Dieter Schmersau, Recurrence equations and their classical orthogonal polynomial solutions, Appl. Math. Comput. 128 (2002), no. 2-3, 303–327. Orthogonal systems and applications. MR 1891025, DOI 10.1016/S0096-3003(01)00078-9
András Kroó, On the exact constant in the $L_2$ Markov inequality, J. Approx. Theory 151 (2008), no. 2, 208–211. MR 2407867, DOI 10.1016/j.jat.2007.09.006
K. H. Kwon and D. W. Lee, Markov-Bernstein type inequalities for polynomials, Bull. Korean Math. Soc. 36 (1999), no. 1, 63–78. MR 1675843
Š. E. Mikeladze, Numerical integration, Uspehi Matem. Nauk (N.S.) 3 (1948), 3–88 (Russian). MR 29282
L. Mirsky, An inequality of the Markov-Bernstein type for polynomials, SIAM J. Math. Anal. 14 (1983), no. 5, 1004–1008. MR 711180, DOI 10.1137/0514079
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Klassicheskie ortogonal′nye polinomy diskretnoĭ peremennoĭ, “Nauka”, Moscow, 1985 (Russian). Edited and with a preface by M. I. Graev. MR 806762
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. Translated from the Russian. MR 1149380, DOI 10.1007/978-3-642-74748-9
Arnold F. Nikiforov and Vasilii B. Uvarov, Special functions of mathematical physics, Birkhäuser Verlag, Basel, 1988. A unified introduction with applications; Translated from the Russian and with a preface by Ralph P. Boas; With a foreword by A. A. Samarskiĭ. MR 922041, DOI 10.1007/978-1-4757-1595-8
Przemysław Rutka and Ryszard Smarzewski, Complete solution of the electrostatic equilibrium problem for classical weights, Appl. Math. Comput. 218 (2012), no. 10, 6027–6037. MR 2873079, DOI 10.1016/j.amc.2011.11.084
Przemysław Rutka and Ryszard Smarzewski, Explicit barycentric formulae for osculatory interpolation at roots of classical orthogonal polynomials, Math. Comp. 86 (2017), no. 307, 2409–2427. MR 3647964, DOI 10.1090/mcom/3184
P. Rutka and R. Smarzewski, The Szegő–Markov–Bernstein inequalities and barycentric representations of the osculatory interpolating operators for classical iterated weights, submitted.
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI, 1975. MR 372517
A. M. Voloschenko and V. I. Zhuravlev, Vychislenie vesov i uzlov kvadraturnyh formul Gaussa s vesovymi funkciyami klassicheskih ortogonal’nyh polinomov nepreryvnogo i diskretnogo peremennogo, Keldysh Institute of Applied Mathematics Preprint No. 89, Moscow, 1977.
Haiyong Wang, Daan Huybrechs, and Stefan Vandewalle, Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials, Math. Comp. 83 (2014), no. 290, 2893–2914. MR 3246814, DOI 10.1090/S0025-5718-2014-02821-4
Haiyong Wang and Shuhuang Xiang, On the convergence rates of Legendre approximation, Math. Comp. 81 (2012), no. 278, 861–877. MR 2869040, DOI 10.1090/S0025-5718-2011-02549-4