Some Researches On Multivariate Graded Hermite Interpolation Problem

The constitution theory of a properly posed set of interpolation function for the multivariate polynomial graded interpolation is studied deeply in the paper. On the basis of Hermite interpolation which along the algebraic curve without multiple factors,we give the approach of graded Hermite interpolation which along the algebraic curve without multiple factors. Puthermore, we give a basic method to construct the graded Hermite interpolation in R~2. In addition,using strong H-basis and r-th order ideal method which is a new mathematic concept and the theory for constructing the properly posed set of interpolation function for interpolation on plane algebraic curve,we give the method to construct the properly posed set of interpolation function for graded interpolation on plane algebraic curve accordingly. At the same time, using the results of ideal and bunch in algebraic geometry ,we study the geometrical structure of properly posed set of interpolation function for graded interpolation on algebraic surface, more over, we make clear the geometrical structure of properly posed set of interpolation function for multivariate graded interpolation basically. On the other hand,when we study the construction of interpolation function ,we introduce a important theorem(G C condition)which is advanced by chung and yao in 1977 and explain the different between it and the method of adding lines which is advanced by professor Liangin 1965 .