On the Leibniz formula for divided differences
DOI:
https://doi.org/10.33993/jnaat351-1011Keywords:
interpolation, divided difference, Lagrange polynomial, Hermite interpolating polynomialAbstract
We give an identity for the Hermite-Lagrange interpolating polynomial and a short proof of Leibniz-type formula for divided differences in case of coalescing knots.Downloads
References
Ampère, A.-M., Essai sur un nouveau mode d'exposition des principes du calcul différentiel, du calcul aux différences et de l'interpolation des suites, considérées comme dérivant d'une source commune, Ann. Math. Pures Appl. (Gergonne), 16, pp. 329-349, 1825.
de Boor, C., A practical guide to splines, Springer-Verlag, New York Heidelberg Berlin, 1978. DOI: https://doi.org/10.1007/978-1-4612-6333-3
de Boor, C., A Leibniz formula for multivariate divided differences, SIAM J. Numer. Anal., 41 (3), pp. 856-868, 2003, https://doi.org/10.1137/s0036142902406818 DOI: https://doi.org/10.1137/S0036142902406818
de Boor, C., Divided differences, Surveys in Approximation Theory, 1, pp. 46-69, 2005.
de Morgan, A., The Differential and Integral Calculus, Baldwin & Cradock, London, 1842.
DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Springer-Verlag, Berlin Heidelberg New York, 1993. DOI: https://doi.org/10.1007/978-3-662-02888-9
Hermite, C., Sur la formule d'interpolation de Lagrange, J. Reine Angew. Math., 84, pp. 70-79, 1878, https://doi.org/10.1515/crll.1878.84.70 DOI: https://doi.org/10.1515/crelle-1878-18788405
Meijering, E., A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing., Proceedings of the IEEE, 90 (3), pp. 319-342, 2002, https://doi.org/10.1109/5.993400 DOI: https://doi.org/10.1109/5.993400
Newton, I., Philosophiae Naturalis Principia Mathematica, Printed by Joseph Streater by order of the Royal Society, London, 1687, https://doi.org/10.5479/sil.52126.39088015628399 DOI: https://doi.org/10.5479/sil.52126.39088015628399
Nicolescu, M., Pic, G., Ionescu, D., Gergely, E., Németi, L., Bal, L. and Radó, F., The mathematical activity of Professor Tiberiu Popoviciu, Studii şi cerc. de matematică (Cluj), 8 (1-2), pp. 7-19, 1957.
Norlund, N. E., Leçons sur les séries d'interpolation, Gauthier-Villars et C^{ie}, Paris, 1926.
Popoviciu, T., Sur quelques propriétés des fonctions d'une ou de deux variables réelles", Ph.D. thesis, Faculté des Sciences de Paris, 1933, published by Institutul de Arte Grafice "Ardealul" (Cluj, Romania).
Popoviciu, T., Introduction à la théorie des différences divisées, Bull. Math. Soc. Roumaine Sci., 42 (1), pp. 65-78, 1940.
Steffensen, J., Note on divided differences, Danske Vid. Selsk. Math.-Fys. Medd., 17 (3), pp. 1-12, 1939.
Waring, E., Problems concerning interpolations, Philosophical Transactions of the Royal Society of London, 69, pp. 59-67, 1779, https://doi.org/10.1098/rstl.1779.0008 DOI: https://doi.org/10.1098/rstl.1779.0008
Whittaker, E. T. and Robinson, G., The Calculus of Observations, Blackie & Son, Limited, London; Glasgow; Bombay, 1924.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.