Taylorian points of an algebraic curve and bivariate Hermite interpolation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, p. 545-577
We introduce and study the notion of Taylorian points of algebraic curves in ${ℂ}^{2}$, which enables us to define intrinsic Taylor interpolation polynomials on curves. These polynomials in turn lead to the construction of a well-behaved Hermitian scheme on curves, of which we give several examples. We show that such Hermitian schemes can be collected to obtain Hermitian bivariate polynomial interpolation schemes.
Classification:  41A05,  41A63,  46A32,  14Q05
@article{ASNSP_2008_5_7_3_545_0,
author = {Bos, Len and Calvi, Jean-Paul},
title = {Taylorian points of an algebraic curve and bivariate Hermite interpolation},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {3},
year = {2008},
pages = {545-577},
zbl = {1177.41001},
mrnumber = {2466439},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_3_545_0}
}

Bos, Len; Calvi, Jean-Paul. Taylorian points of an algebraic curve and bivariate Hermite interpolation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, pp. 545-577. http://www.numdam.org/item/ASNSP_2008_5_7_3_545_0/

[1] B. Bojanov and Y. Xu., On polynomial interpolation of two variables, J. Approx. Theory 120 (2003), 267-282. | MR 1959868 | Zbl 1029.41001

[2] B. D. Bojanov, H. A. Hakopian and A. A. Sahakian, “Spline Functions and Multivariate Interpolations”, Mathematics and its Applications, Vol. 248, Academic Publishers Group, Dordrecht, 1993. | MR 1244800 | Zbl 0772.41011

[3] L. Bos, On certain configurations of points in ${𝐑}^{n}$ which are unisolvent for polynomial interpolation, J. Approx. Theory 64 (1991), 271-280. | MR 1094439 | Zbl 0737.41002

[4] L. P. Bos and J.-P. Calvi, Multipoint taylor interpolation, Calcolo 51 (2008), 35-51. | MR 2400160 | Zbl 1168.41301

[5] J.-P. Calvi and L. Filipsson, The polynomial projectors that preserve homogeneous differential relations: a new characterization of Kergin interpolation, East J. Approx. 10 (2004), 441-454. | MR 2101102 | Zbl 1113.41002

[6] D. Cox, J. Little and D. O'Shea, “Ideals, Varieties, and Algorithms”, Undergraduate Texts in Mathematics, Springer, New York, third edition, 2007. | MR 2290010 | Zbl 1118.13001

[7] C. De Boor and A. Ron, On multivariate polynomial interpolation, Constr. Approx. 6 (1990), 287-302. | MR 1054756 | Zbl 0719.41006

[8] C. De Boor and A. Ron, The least solution for the polynomial interpolation problem, Math. Z. 210 (1992), 347-378. | MR 1171179 | Zbl 0735.41001

[9] Lars Filipsson, Complex mean-value interpolation and approximation of holomorphic functions, J. Approx. Theory 91 (1997), 244-278. | MR 1484043 | Zbl 0904.32012

[10] M. Gasca and T. Sauer, Polynomial interpolation in several variables, Adv. Comput. Math. 12 (2000), 377-410. Multivariate polynomial interpolation. | MR 1768957 | Zbl 0943.41001

[11] H. A. Hakopian and M. F. Khalaf, On the poisedness of Bojanov-Xu interpolation, J. Approx. Theory 135 (2005), 176-202. | MR 2158529 | Zbl 1076.41002

[12] H. A. Hakopian and M. F. Khalaf, On the poisedness of Bojanov-Xu interpolation, II, East J. Approx. 11 (2005), 187-220. | MR 2151615 | Zbl 1247.46025

[13] F. Kirwan, “Complex Algebraic Curves”, London Mathematical Society Student Texts, Vol. 23, Cambridge University Press, Cambridge, 1992. | MR 1159092 | Zbl 0744.14018

[14] R. A. Lorentz, “Multivariate Birkhoff Interpolation”, Lecture Notes in Mathematics, Vol. 1516. ix, Springer-Verlag, 1992. | MR 1222648 | Zbl 0760.41002

[15] R. A. Lorentz, Multivariate Hermite interpolation by algebraic polynomials: A survey, J. Comput. Appl. Math. 122 (2000), 167-201. | MR 1794655 | Zbl 0967.65008

[16] M. G. Marinari, H. M. Möller and T. Mora, Gröbner bases of ideals defined by functionals with an application to ideals of projective points, Appl. Algebra Engrg. Comm. Comput. 4 (1993), 103-145. | MR 1223853 | Zbl 0785.13009

[17] H. M. Möller, Hermite interpolation in several variables using ideal-theoretic methods, In: “Constructive Theory of Functions of Several Variables”, Proc. Conf., Math. Res. Inst., Oberwolfach, 1976, Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, 155-163. | MR 493046 | Zbl 0347.41002