Multivariate Complete Interpolation Basis And Multivariate Rational Interpolation Of Zero Degree

Let V = {x⁽⁰⁾,…, x^(m-1)} be a finite set of m distinct nodes in R^d. For each node x⁽ⁱ⁾ an ordinate f_i∈Ris given. The problem of d—variate Lagrange interpolation is to find a suitable polynomial p∈R[x₁,…, x_d] that takes these prescribed m values, where R[x₁,…, x_d] is the ring of all d—variate polynomials with real coefficients. That is, we wantWhen this occurs, we call p an interpolation polynomial of interpolation problem and write it in the formwithω₀,…,ω_m-1 d—variate monomials.Well known, differing thoroughly from univariate Lagrange interpolation, the structure of a multivariate Lagrange interpolation polynomial depends not only on the number, but considerably on the geometry of the interpolation sites [51]. With constructive algebraic geometry tools, one can establish the algebraic theory of multivariate Lagrange interpolation directly.If the nodes in V are in general position and number m satisfies[50] had proved that the MDIMB for Lagrange interpolation w.r.t. (?) consists of all monomials in T_n^d. However, when the nodes are not in general position, we want to know that under which condition can all monomials in T_n'^d, for some n' < n, be contained in the the MDIMB.Lemma 1 Let U (?) V be two finite sets of distinct nodes in R^d. Then the minimal degree interpolation monomial basis for Lagrange interpolation on U w.r.t. (?) is a subset of the one on V.Theorem 1 Given a finite set V = {x⁽⁰⁾, x⁽¹⁾,…, x^(m-1)} (?) R^d of m distinct nodes. If there exist a subset of nodes U (?) V and a nonnegative integer n∈N such that and the nodes in U are not all located on any algebraic manifold of degree at most n, then the MDIMB for Lagrange interpolation on V w.r.t. (?) must consist of T_n^d.The corollary in the following is straightforward.Corollary 1 There exists a maximum n_v∈N such that the MDIMB for Lagrange interpolation on V w.r.t. (?) consists of T_nV^d. We will call this T_nV^d the maximal complete Lagrange interpolation basis w.r.t. V.Theorem 2 If N_V is the MDIMB for Lagrange interpolation on V w.r.t. (?), andωis highest one of N_V refer to (?). Then there must be one point (?) in V, which leads to the N_U = N_V\{ω} will be MDIMB for Lagrange interpolation on w.r.t. (?)Corollary 2 Fix graded lexicographic order (?), for and limited notes V (?) Rⁿ, N_V is the MDIMB for Lagrange interpolation on V w.r.t. (?) and N_V (?) T_n^d.Then there must be a subsetU of V,s.t. N_U = T_n^d, here N_U is the MDIMB for Lagrange interpolation on U w.r.t. (?)Corollary 3 Let V = {x⁽⁰⁾, ... , x^(m-1)} be a finite set of m distinct nodes in R^d, N_V = {ω₀,ω₁,…,ω_m-1}.ω₀(?)ω_<sub>1 (?)…<ω_m-1} in a certain order (?). Then we can reorder the points in V, x⁽⁰⁾, x⁽¹⁾,…, x^(m-1) such that V_k = {x⁽⁰⁾, x⁽¹⁾,…, x^(k-1)} for all 0V_k = {ω₀,ω₁,…,ω_k-1}. We will define Nodes in the order of (?) in this condition.Theorem 3 If N_V is the MDIMB for Lagrange interpolation on V w.r.t. (?), then there exists a subset U (?) V andW (?) U such that the MDIMB N_U for Lagrange interpolation on U w.r.t. (?) is contained in N_U and a nonlacunary interpolation basis, N_W for Lagrange interpolation on W w.r.t. (?) is contained in N_W and a complete interpolation basis. We have a algorithm for finding all of the subsets of V on which Lagrange interpolation basis is complete.Let V = {x⁽⁰⁾,…, x^(m+n)} be a finite set of m + n + 1 distinct nodes in R^d. For given values f₀,…, f_m+n∈R, seek d—variate rational interpolation functionso thatThe interpolation problem is called a d—variate rational interpolation(of Cauchy type).Apparently, when q is a nonzero constant (polynomial of zero degree), the rational interpolation problem corresponding to Eq. degenerates to a Lagrange interpolation problem. With a multivariate polynomial interpolation algorithm such as the one in , we can work out the MDIMB, say {ω₀ = 1,ω₁,…,ω_m+n}, for Lagrange interpolation on V w.r.t. (?), withω_i (?)ω_i+1, i = 0,…, m + n - 1. Use similar method in the theory of univariate rational interpolation, we takeThen the rational interpolation problem is referred to as a rational interpolation of (m, n) type.The lemma in the following presents not only the existence of multivariate rational interpolation of (m, n) type, but also the way of constructing the interpolation function.Then we have this lemma:Lemma 2 [52] Given a finite set V = {x⁽⁰⁾,…, x^(m+n)} of distinct nodes in R^d. Assume that the MDIMB for Lagrange interpolation on V w.r.t. (?) is where . Ifwhereω_i^(l):=ω_i(x^(l)), f₀,…, f_m+n∈R are given real numbers, then rational function R = p/q satisfies this Eq. if and only if square matrixes B_j(x^(j)), j = 0,…, m + n, of order m + n are nonsingular, whereApply this lemma , we can deduce the following main theorem immediately.Theorem 4 Let V = {x⁽⁰⁾,…, x^(m+n)} be a set of m + n + 1 distinct nodes in R^d. Suppose U is a subset of V with such that the MDIMB for Lagrange interpolation on U is T_k^d, where k|- is the greatest one that satisfies this Eq. . Assume that U = {x^(i₀),…, x^(i_N)}, , where N = #U - 1, ,j = 0,…, N - 1. For any subset W = U∪{x⁽(i_N+1)),…, x⁽(i_2N))}, where x^(i_j)∈V\U, j = N + 1,…, 2N, ifwhereω_s^(l) :=ω_s(x^(i_l)), f₀,…, f_2N∈R are given values, then in order that rational function of zero degree R = p/q satisfyit is necessary and sufficient that square matrixes B_j(x^(i_j)) of order 2N, j = 0,…,2N, be nonsingular, whereFrom that theorem , we modify some parts of Algorithm to get the following algorithm for constructing node subsets of V suitable for multivariate rational interpolation of zero degree.