Abstract
We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the results states that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for each monomial order there is a unique polynomial that interpolates on the points in the variety. The result is motivated by the problem of constructing cubature formulae, and it leads to a theorem on cubature formulae which can be considered an extension of Gaussian quadrature formulae to several variables.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Bojanov, H.A. Hakopian and A.A. Sahakian, Spline Functions and Multivariate Interpolations Mathematics and its Applications, Vol. 248 (Kluwer Academic, Dordrecht, 1993).
D. Cox, J. Little and D. O'shea, Ideals, Varieties, and Algorithms, 2nd ed. (Springer, Berlin, 1997).
C. de Boor and A. Ron, On multivariate polynomial interpolation, Constr. Approx. 6 (1990) 287–302.
C. de Boor and A. Ron, On polynomial ideals of finite codimension with applications to box spline theory, J. Math. Anal. Appl. 158 (1991) 168–193.
C. de Boor and A. Ron, The least solution for the polynomial interpolation problem, Math. Z. 210 (1992) 347–378.
M. Gasca, Multivariate polynomial interpolation, in: Computation of Curves and Surfaces, eds. W. Dahmen, M. Gasca and C.A. Micchelli (Kluwer Academic, Dordrecht, 1990).
T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: Theory and Application of Special Functions, ed. R.A. Askey (Academic Press, New York, 1975) pp. 435–495.
R.A. Lorentz, Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, Vol. 1516 (Springer, Berlin, 1992).
H.M. Möller, Polynomideale und Kubaturformeln, Thesis, University of Dortmund (1973).
H.M. Möller, Kubaturformeln mit minimaler Knotenzahl, Numer. Math. 35 (1976) 185–200.
H.M. Möller, On the construction of cubature formulae with few nodes using Groebner bases, in: Numerical Integration, eds. P. Keast and G. Fairweather (D. Reidel, 1998) pp. 177–192.
I.P. Mysovskikh, Numerical characteristics of orthogonal polynomials in two variables, Vestnik Leningrad Univ. Math. 3 (1976) 323–332.
I.P. Mysovskikh, The approximation of multiple integrals by using interpolatory cubature formulae, in: Quantitative Approximation, eds. R.A. DeVore and K. Scherer (Academic Press, New York, 1980).
I.P. Mysovskikh, Interpolatory Cubature Formulas (Nauka, Moscow, 1981) (in Russian).
Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948) 286–300.
Th. Sauer, Polynomial interpolation of minimal degree, Numer. Math. 78 (1997) 59–85.
Th. Sauer, Polynomial interpolation of minimal degree and Gröbner bases, in: Gröbner Bases and Applications, eds. B. Buchberger and F. Winkler, London Mathematical Society Lecture Notes Series, Vol. 251 (Cambridge Univ. Press, 1998) pp. 483–494.
A. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, NJ, 1971).
Y. Xu, Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature, Pitman Research Notes in Mathematics Series (Longman, Essex, 1994).
Y. Xu, On orthogonal polynomials in several variables, in: Special Functions, q-Series, and Related Topics, Fields Inst. Commun., Vol. 14 (1997) pp. 247–270.
Y. Xu, Cubature formulae and polynomial ideals, Adv. Appl. Math. 23 (1999) 211–233.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Xu, Y. Polynomial interpolation in several variables, cubature formulae, and ideals[*]Supported by the National Science Foundation under Grant DMS-9802265.. Advances in Computational Mathematics 12, 363–376 (2000). https://doi.org/10.1023/A:1018989707569
Issue Date:
DOI: https://doi.org/10.1023/A:1018989707569