Abstract
Introduced by G. Birkhoff and popularized by C. de Boor, ideal projectors are an elegant generalization of Hermite interpolation projectors to the multivariate setting. An important class of ideal projectors comprises Lagrange interpolation projectors. In this article, we give a characterization of Lagrange projectors in terms of their “restriction property.”
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References
Birkhoff, G.: The algebra of multivariate interpolation. In: Constructive Approaches to Mathematical Models (Proc. Conf. in honor of Duffin, R.J., Pittsburgh, Pa., 1978), pp. 345–363. Academic Press, New York (1979)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2007)
de Boor, C., Ron, A.: On polynomial ideals of finite codimension with applications to box spline theory. J. Math. Anal. Appl. 158(1), 168–193 (1991). http://dx.doi.org/10.1016/0022-247X(91), 90275–5
de Boor, C.: Ideal interpolation. In: Approximation Theory XI: Gatlinburg 2004, Mod. Methods Math., pp. 59–91. Nashboro Press, Brentwood, TN (2005)
Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Revised reprint of the 1916 original, with an introduction by Paul Roberts. Cambridge University Press, Cambridge (1994)
Möller, H.M.: Hermite interpolation in several variables using ideal-theoretic methods. In: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), pp. 155–163. Lecture Notes in Math., vol. 571. Springer, Berlin (1977)
Sauer, T., Xu, Y.: On multivariate Lagrange interpolation. Math. Comp. 64(211), 1147–1170 (1995). http://dx.doi.org/10.2307/2153487
Sauer, T.: Polynomial interpolation in several variables: lattices, differences, and ideals. In: Topics in Multivariate Approximation and Interpolation, Stud. Comput. Math., vol. 12, pp. 191–230. Elsevier, Amsterdam (2006). http://dx.doi.org/10.1016/S1570-579X(06), 80009–1
Shekhtman, B.: A taste of ideal interpolation. J. Concr. Appl. Math. 8(1), 125–149 (2010)
Acknowledgments
I would like to thank Carl de Boor for his many contributions to this paper; both in substance and style.
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Shekhtman, B. (2014). One Characterization of Lagrange Projectors. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_19
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DOI: https://doi.org/10.1007/978-3-319-06404-8_19
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