Summary
We try to solve the bivariate interpolation problem (1.3) for polynomials (1.1), whereS is a lower set of lattice points, and for theq-th interpolation knot,A q is the set of orders of derivatives that appear in (1.3). The number of coefficients |S| is equal to the number of equations Σ|A q |. If this is possible for all knots in general position, the problem is almost always solvable (=a.a.s.). We seek to determine whether (1.3) is a.a.s. An algorithm is given which often gives a positive answer to this. It can be applied to the solution of a problem of Hirschowitz in Algebraic Geometry. We prove that for Hermite conditions (1.3) (when allA q are lower triangles of orderp) andP is of total degreen, (1.3) is a.a.s. for allp=1, 2, 3 and alln, except for the two casesp=1,n=2 andp=1,n=4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chudnovsky, G.V.: Singular points on complex hypersurfaces and multidimensional Schwarz lemma. Seminar on Number Theory, Paris 1979–89, pp. 29–69. Progr. Math., Vol. 12. Boston, MA: Birkhäuser 1981
Hakopian, H.A.: Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type. J. Approximation Theory34, 286–305 (1982)
Hirschowitz, A.: La méthode d'Horace pour l'interpolation à plusieurs variables. Manuser. Math.50, 337–388 (1985)
Lorentz, G.G.: Solvability of multivariate interpolation. J. Reine Angew. Math. (to appear)
Lorentz, G.G., Lorentz, R.A.: Multivariate interpolation. In: Rational approximation and interpolation, Lecture Notes in Mathematics, vol. 1105, pp. 136–144. Berlin Heidelberg New York: Springer 1984
Lorentz, G.G., Lorentz, R.A.: Solvability problems of bivariate interpolation. I. Constructive Approximation2, 153–170 (1986)
Lorentz, G.G., Lorentz, R.A.: Solvability of bivariate approximation, II. Applications. Approximation Theory Appl.3, 79–97 (1987)
Lorentz, R.A.: Uniform bivariate Hermite interpolation I. Coordinate degree. preprint
Micchelli, C.A.: A constructive approach to Kergin interpolation inR k: MultivariateB-splines and Lagrange interpolation. Rocky Mt. J. Math.10, 485–497 (1980)
Mordell, L.J.: Diophantine equations. London: Academic Press 1969
Author information
Authors and Affiliations
Additional information
Dedicated to R. S. Varga on the occasion of his sixtieth birthday
This work has been partly supported by the Texas ARP and the Deutsche Forschungsgemeinschaft
Rights and permissions
About this article
Cite this article
Lorentz, G.G., Lorentz, R.A. Bivariate hermite interpolation and applications to Algebraic Geometry. Numer. Math. 57, 669–680 (1990). https://doi.org/10.1007/BF01386436
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01386436