Studies On The Regularity Of Multivariate Hermite Interpolation Of Type Total Degree

With the development of information technology and the need of science and engineering calculation,multivariate interpolation,especially multivariate polynomial interpolation,is be-coming more and more important.It has been widely used in many fields,such as surfaces design,finite element method and so on.However,the multivariate polynomial interpolation problem is more complex than the univariate case.Even if the number of interpolation nodes is equal to the dimension of interpolation space,the existence and uniqueness of the interpola-tion polynomial can not be guaranteed.Hence,we must resolve the poisedness of multivariate interpolation problem first of all.In this paper,we study the poisedness of multivariate Hermite interpolation problem of type total degree.One is on m nodes with m?1/2d(d + 3)and the other is on 2d nodes.The main work is shown as follows:(1)Suppose m ?1/2d(d + 3)and p1?p2?…?Pm.We first check the degree of interpolation polynomial and the results show that only if n=pm+pm-1+1 interpolation problem is possible to be well posed.Then,we consider the solutions of diophantine equation only in three cases.In each case,we discuss the solutions of diophantine equation and obtain a series of interpolation schemes.We prove the singularity theoretically and the almost regularity by numerical method.One can choose a set of nodes and calculate the Vandermonde determi-nant.If it is non-zero,then the interpolation scheme is almost regular.Especially,we present theoretical proofs for three families of almost regular schemes.But,the well-posedness of some schemes are left open.Nevertheless,the method of solving the above diophantine equation and the proof method of the singularity or almost regularity are still valid for m>1/2d(d + 3).(2)For any d ? 2,we provide a class of almost regular interpolation schemes.When m =2d.two combinatorial identities are given,which are two families of solutions of the above diophantine equation.Then the almost regularity of the corresponding multivariate Hermite interpolation problem of type total degree is proved and obtain two families of almost regular interpolation schemes.Based on the two families of interpolation schemes,we get d-2 almost regular interpolation schemes and the number of nodes is(?),k = 2,3,…,d-1 respectively.That is,for any d ? 2,we obtain d almost regular interpolation schemes.In the end,we present a constructive proof of almost regularity of this kind of interpolation scheme and this proof method can be used to construct almost regular interpolation schemes in high dimensional space by the almost regular ones in low dimensional space.