On the Leibniz formula for divided differences

Mircea Ivan

doi:10.33993/jnaat351-1011

Authors

Mircea Ivan Technical University of Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat351-1011

Keywords:

interpolation, divided difference, Lagrange polynomial, Hermite interpolating polynomial

Abstract views: 508

Abstract

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

Download data is not yet available.

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.