The Research On Multivariate Interpolation And Dimension Of Interpolation Space

Content: Interpolation is a very classic problem of Mathematics andalso a basic problem in computational Mathematics. It is well knownthat univariate interpolation has a very well developed theory andmethod. However, since 1980's, people come to turn the research ofinterpolation on multivariate interpolation, the reason is that multi-variate interpolation has a widespread application in many fields, suchas surfaces design, multivariate function arrange figures and finite ele-ment method. At the same time, since the algebraic geometry theoryand method during the past few years have unceasingly developed andimproved, we have forceful theory basis and new method to further re-search the problem of multivariate interpolation. In this paper, thereare four sections. In the first section, we introduce the basic theory. Inthe second section, we give out the recent main results of multivariateLagrange interpolation. In the third section, we get some theories ofno posed set of nodes from a kind of recursive method of constructingposed set of nodes for bivariate Lagrange interpolation, and we get akind of abnormal posed set of nodes for interpolation form the reversethought of that theory. In the four section, from the interpolationdimension of an algebraic curve and an algebraic surface,using theoryand method of algebraic geometry, we get some interpolation of analgebraic curve and an algebraic surface, and we prove the reasonableof the interpolation dimension of an algebraic curve and an algebraicsurface.