×

Two chain rules for divided differences and Faá di Bruno’s formula. (English) Zbl 1113.05011

Faà di Bruno’s formula is known to express a derivative of arbitrary order of a composite function \(g=f\circ \varphi\) in terms of derivatives of \(f\) and \(\varphi\). The authors prove two explicit formulae that express an arbitrary divided difference of a composite function \(g=f\circ \varphi\) in terms of divided differences of \(f\) and \(\varphi \). Both formulae provide a simple proof of Faà di Bruno’s formula.

MSC:

05A17 Combinatorial aspects of partitions of integers
05A18 Partitions of sets
26A06 One-variable calculus
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
41A05 Interpolation in approximation theory
65D05 Numerical interpolation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carl de Boor, Divided differences, Surv. Approx. Theory 1 (2005), 46 – 69. · Zbl 1071.65027
[2] Nira Dyn and Charles A. Micchelli, Piecewise polynomial spaces and geometric continuity of curves, Numer. Math. 54 (1988), no. 3, 319 – 337. · Zbl 0638.65010 · doi:10.1007/BF01396765
[3] C. F. Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math 1 (1857), 359-360.
[4] M. S. Floater, Arc length estimation and the convergence of parametric polynomial interpolation, BIT 45 (2005), 679-694. · Zbl 1095.65011
[5] John A. Gregory, Geometric continuity, Mathematical methods in computer aided geometric design (Oslo, 1988) Academic Press, Boston, MA, 1989, pp. 353 – 371.
[6] T. N. T. Goodman, Properties of \?-splines, J. Approx. Theory 44 (1985), no. 2, 132 – 153. · Zbl 0569.41010 · doi:10.1016/0021-9045(85)90076-0
[7] Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966.
[8] Warren P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217 – 234. · Zbl 1024.01010 · doi:10.2307/2695352
[9] Charles Jordan, Calculus of finite differences, Third Edition. Introduction by Harry C. Carver, Chelsea Publishing Co., New York, 1965. · Zbl 0060.12309
[10] Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0895.68055
[11] Knut Mørken and Karl Scherer, A general framework for high-accuracy parametric interpolation, Math. Comp. 66 (1997), no. 217, 237 – 260. · Zbl 0854.41001
[12] Elena Popoviciu, Sur quelques propriétés des fonctions quasi-convexes, Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1983) Preprint, vol. 83, Univ. ”Babeş-Bolyai”, Cluj-Napoca, 1983, pp. 107 – 114 (French). · Zbl 0581.26012
[13] Tiberiu Popoviciu, Introduction à la théorie des différences divisées, Bull. Math. Soc. Roumaine Sci. 42 (1940), no. 1, 65 – 78 (French). · JFM 66.0389.01
[14] John Riordan, Derivatives of composite functions, Bull. Amer. Math. Soc. 52 (1946), 664 – 667. · Zbl 0063.06505
[15] John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. · Zbl 0078.00805
[16] Steven Roman, The formula of Faà di Bruno, Amer. Math. Monthly 87 (1980), no. 10, 805 – 809. · Zbl 0513.05009 · doi:10.2307/2320788
[17] J. F. Steffensen, Interpolation, Baltimore, 1927. · JFM 53.0524.01
[18] J. F. Steffensen, Note on divided differences, Danske Vid. Selsk. Math.-Fys. Medd 17 (1939), 1-12.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.