The properly posed set of nodes of multivariate Lagrange interpolation along algebraic manifold

Abstract

In this article, Lagrange interpolation by polynomials in several variables is studied in a systematic way. After defining properly posed set of nodes (PPSN) along the sufficiently intersected algebraic manifolds, we give the sufficient and necessary condition for judging PPSN and propose the generally constructive method of PPSN for Lagrange interpolation along algebraic manifolds, which is named as an algebraic hypersurface-superposition process. The theory is illustrated with a simple example in ℝ³. Moreover, the relationship between the sufficiently intersected polynomials and H-basis is also discussed.

Keywords:

• multivariate Lagrange interpolation
• PPSN
• algebraic manifold

AMS Subject Classifications:

• 41A05
• 65D05

Acknowledgements

The authors would like to thank Prof Zhongxuan Luo of Dalian University of Technology for his valuable comments and suggestions which helped to improve the article. This project was supported by the National Nature Science Foundation of China (No. 10771028, No. 10801023).