On the history of multivariate polynomial interpolation

Journal of Computational and Applied Mathematics

Volumn 122, Issue 1, 2000, Pages 23-35

  Gasca, Mariano ^a   Sauer, Thomas ^b  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; FINITE ELEMENT METHOD; POLYNOMIALS; TENSORS;

HERMITE INTERPOLATION; MULTIVARIATE POLYNOMIAL INTERPOLATION;

INTERPOLATION;

MATHEMATICAL METHOD;

EID: 0034289836     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00353-8     Document Type: Article

References (53)

1. A.C. Ahlin, A bivariate generalization of Hermite's interpolation formula, Math. Comput. 18 (1964) 264-273.
2. I.S. Berezin, N.P. Zhidkov, Computing Methods, Addison-Wesley, Reading, MA, 1965 (Russian version in 1959).
3. O. Biermann, Über näherungsweise Kubaturen, Monatshefte Math. Phys. 14 (1903) 211-225.
4. O. Biermann, Vorlesungen über Mathematische Näherungsmethoden, Vieweg, Braunschweig, 1905.
5. G. Birkhoff, M.H. Schultz, R.S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations, Numer. Math. 11 (1968) 232-256.
6. W. Borchardt, Über eine Interpolationsformel für eine Art symmetrischer Funktionen und deren Anwendung, Abh. d. Preuß. Akad. d. Wiss. (1860) 1-20.
7. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
8. n with applications to finite element methods, Arch. Rational Mech. Anal. 46 (1972) 178-199.
9. P.G. Ciarlet, C. Wagschal, Multipoint Taylor formulas and applications to the finite element method, Numer. Math. 17 (1971) 84-100.
10. C. Coatmelec, Approximation et interpolation des fonctions differentiables de plusieurs variables, Ann. Sci. Ecole Norm. Sup. 83 (1966) 271-341.
11. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943) 1-23.
12. P.J. Davis, Interpolation and Approximation, Blaisdell, Walthan, MA, 1963 (2nd Edition, Dover, New York, 1975).
13. Encyklopädie der mathematischen Wissenschaften, Teubner, Leipzig, pp. 1900-1904.
14. Encyclopédie des Sciences Mathematiques, Gauthier-Villars, Paris, 1906.
15. I.A. Ezrohi, General forms of the remainder terms of linear formulas in multidimensional approximate analysis I, II Mat. Sb. 38 (1956) 389-416 and 43 (1957) 9-28 (in Russian).
16. M. Gasca, T. Sauer, Multivariate polynomial interpolation, Adv. Comput. Math., 12 (2000) 377-410.
17. G. Glaeser, L'interpolation des fonctions differentiables de plusieurs variables, in: C.T.C. Wall (Ed.), Proceedings of Liverpool Singularities Symposium II, Lectures Notes in Mathematics, Vol. 209, Springer, Berlin, 1971, pp. 1-29.
18. H.H. Goldstine, A History of Numerical Analysis from the 16th Through the 19th Century, Springer, Berlin, 1977.
19. R.B. Guenter, E.L. Roetman, Some observations on interpolation in higher dimensions, Math. Comput. 24 (1970) 517-521.
20. E. Isaacson, H.B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966.
21. C.G.J. Jacobi, Theoremata nova algebraica circa systema duarum aequationum inter duas variabiles propositarum, Crelle J. Reine Angew. Math. 14 (1835) 281-288.
22. L. Kronecker, Über einige Interpolationsformeln für ganze Funktionen mehrerer Variabeln. Lecture at the academy of sciences, December 21, 1865, in: H. Hensel (Ed.), L. Kroneckers Werke, Vol. I, Teubner, Stuttgart, 1895, pp. 133-141. (reprinted by Chelsea, New York, 1968).
23. K.S. Kunz, Numerical Analysis, McGraw-Hill, New York, 1957.
24. G.G. Lorentz, Approximation of Funtions, Chelsea, New York, 1966.
25. R. Lorentz, Multivariate Hermite interpolation by algebaic polynomials: a survey, J. Comput. Appl. Math. (2000) this volume.
26. S.E. Mikeladze, Numerical Methods of Mathematical Analysis, Translated from Russian, Office of Tech. Services, Department of Commerce, Washington DC, pp. 521-531.
27. S. Narumi, Some formulas in the theory of interpolation of many independent variables, Tohoku Math. J. 18 (1920) 309-321.
28. L. Neder, Interpolationsformeln für Funktionene mehrerer Argumente, Skandinavisk Aktuarietidskrift (1926) 59.
29. R.A. Nicolaides, On a class of finite elements generated by Lagrange interpolation, SIAM J. Numer. Anal. 9 (1972) 435-445.
30. R.A. Nicolaides, On a class of finite elements generated by Lagrange interpolation II, SIAM J. Numer. Anal. 10 (1973) 182-189.
31. K. Pearson, On the construction of tables and on interpolation, Vol. 2, Cambridge University Press, Cambridge, 1920.
32. J. Radon, Zur mechanischen Kubatur, Monatshefte Math. Physik 52 (1948) 286-300.
33. H.E. Salzer, Note on interpolation for a function of several variables, Bull. AMS 51 (1945) 279-280.
34. H.E. Salzer, Table of coefficients for interpolting in functions of two variables, J. Math. Phys. 26 (1948) 294-305.
35. H.E. Salzer, Note on multivariate interpolation for unequally spaced arguments with an application to double summation, J. SIAM 5 (1957) 254-262.
36. H.E. Salzer, Some new divided difference algorithms for two variables, in: R.E. Langer (Ed.), On Numerical Approximation, University of Wisconsin Press, Madison, 1959, pp. 61-98.
37. H.E. Salzer, Divided differences for functions of two variables for irregularly spaced arguments, Numer. Math. 6 (1964) 68-77.
38. H.E. Salzer, Formulas for bivariate hyperosculatory interpolation, Math. Comput. 25 (1971) 119-133.
39. H.E. Salzer, G.M. Kimbro, Tables for Bivariate Osculatory Interpolation over a Cartesian Grid, Convair Astronautics, 1958.
40. A. Sard, Remainders: functions of several variables, Acta Math. 84 (1951) 319-346.
41. A. Sard, Remainders as integrals of partial derivatives, Proc. Amer. Math. Soc. 3 (1952) 732-741.
42. T. Sauer, Yuan Xu, On multivariate Lagrange interpolation, Math. Comput. 64 (1995) 1147-1170.
43. T. Sauer, Yuan Xu, A case study in multivariate Lagrange interpolation, in: S.P. Singh (Ed.), Approximation Theory, Wavelets and Applications, Kluwer Academic Publishers, Dordrecht, 1995, pp. 443-452.
44. D.D. Stancu, The remainder of certain linear approximation formulas in two variables, J. SIAM Numer. Anal. 1 (1964) 137-163.
45. I.F. Steffensen, Interpolation, Chelsea, New York, 1927 (2nd Edition, 1950).
46. G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
47. H.C. Thacher Jr., W.E. Milne, Interpolation in several variables, J. SIAM 8 (1960) 33-42.
48. H.C. Thacher Jr., Derivation of interpolation formulas in several independent variables, Ann. N.Y. Acad. Sci. 86 (1960) 758-775.
49. T.N. Thiele, Interpolationsrechnung, Teubner, Leipzig, 1909.
50. R.S. Walker, Algebraic Curves, Springer, Berlin, 1978.
51. E.T. Whittaker, G. Robinson, Calculus of Observations, 4th Edition, Blackie and Sons, London, 1944.
52. O.C. Zienkiewicz, The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967.
53. M. Zlamal, On the finite element method, Numer. Math. 12 (1968) 394-409.