Abstract
In this chapter, as warmup for rational approximation, we discuss polynomial approximation. Since rational approximation has f = P∕Q ⇔ Qf = P, finding P and Q can be viewed as a kind of polynomial approximation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
H. B. Curry and I. J. Schoenberg, On Pólya frequency functions. IV. The fundamental spline functions and their limits, J. Analyse Math. 17 (1966), 71–107.
C. de Boor, Divided Differences, Surveys in Approximation Theory, 1 (2005), 46–69.
C. de Boor, A practical guide to splines, Revised edition, Springer-Verlag, New York, 2001; original: 1978.
N. Eraser, Newton’s Interpolation Formulas, J. Institute Actuaries 51 (1919), 211–232.
M. S. Floater and T. Lyche, Two chain rules for divided differences and Faa di Bruno’s formula, Math. Comp. 76 (2007), 867–877.
O. Heinävaara, Matrix monotone functions, University of Helsinkii Thesis, unpublished; available at https://helda.helsinki.fi/handle/10138/273502.
C. Hermite, Sur la formule d’interpolation de Lagrange, J. Reine Ang. Math. 84 (1878), 70–79.
E. Hopf, Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Universität Berlin dissertation, 1926.
J. L. Lagrange, Leçons élémentaires sur les mathématiques données à l’École Normale en 1795. Séances des Écoles normales an III (1794/95). Reprinted in Journal de l’École Polytechnique 2, 173-278.
I. Newton, Methodus differentialis, Jones, London, 1711.
G. Peano, Resto helle formule di quadratura espresso con un integrale definito, Atti della Reale Accademia dei Lincei, Rendiconti 22 (1913), 562–569.
G. Peano, Residuo in formulas de quadratura, Mathesis 4 (1914), 5–10.
T. Popoviciu, Sur quelques propriétés des fonctions d’une ou de deux variables réelles, dissertation presented to the Faculté des Sciences de Paris (1933), published by Institutul de Arte Grafice “Ardealul” (Cluj, Romania) .
A. Sard, Integral representations of remainders, Duke Math. J. 15 (1948), 333–345.
H. A. Schwarz, Démonstration élémentaire d’une propriété fondamentale des fonctions interpolaires, Atti Accad. Sci. Torino 17 (1881–1882), 740–742.
J. F. Steffensen, Note on divided differences, Danske Vid. Selsk. Math.-Fys. Medd. 17(3) (1939), 1-12.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Simon, B. (2019). Divided Differences and Polynomial Approximation. In: Loewner's Theorem on Monotone Matrix Functions. Grundlehren der mathematischen Wissenschaften, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-22422-6_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-22422-6_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22421-9
Online ISBN: 978-3-030-22422-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)