Some researches on multivariate Lagrange interpolation along the sufficiently intersected algebraic manifold

Abstract

In this article, Lagrange interpolation by polynomials in several variables is studied. Particularly on the sufficiently intersected algebraic manifolds, we discuss the dimension about the interpolation space of polynomials. After defining properly posed set of nodes (or PPSN for short) along the sufficiently intersected algebraic manifolds, we prove the existence of PPSN and give the number of points in PPSN of any degree. Moreover, in order to compute the number of points in PPSN concretely, we propose the operator ∇ _k with reciprocal difference.

Keywords:

• Multivariate Lagrange interpolation
• PPSN
• Algebraic manifold
• Mathematics Subject Classifications
• 41A05
• 65D05

Acknowledgements

The authors would like to thank Professor Zhongxuan Luo of Dalian University of Technology for his valuable comments and suggestions which helped to improve the article. The project was supported by the National Nature Science Foundation of China (Nos. 10526013, 10471018 and 60673021).