Tensor Product Space P <sub> M1, M2, ..., Ms </ Sub> Multivariate Lagrange Interpolation Posed Node Set,

In this paper ,we construct a class of properly posed set of nodes(or PPSN,for short)for multivariate Lagrange interpolation about the interpolation place of Pmi,mi,-,m,(s > 2), after we study the geometric structure of PPSN along an algebraic hyperplane Xj ?a = 0, 1 < 3 < s.ln this way, we can generalize the technique about Pm.n in [2],Definition Let . s be an algebraic hyperplane in Cs,and f relationsa PPSN along the algebraic hyperplane Theorem 1 Let {rJJ=1be a PPSN in Cs about Pmi,m.2,...,mi. which are not on an algebraic hyperplane x} ??a ?0. 1 < j < s. and {gj}f=1be a PPSN along the algebraic hyperplane Xj ?a ?0 about -Pm1,...,mJ_l,mJ+i,m,+1,...,ms, then the set of {ri}|=1U{g!}f=1must be a PPSNin Cs aboutTheorem 2 Let =i be on an algebraic hyperplanes) R is a PPSN along Xj - a = 0 about Pmi,-,mi-l,mi+i,mi+l>-,mt if and only if R is a PPSN of Pmi .....n,^, ,o,m ;.j .....mTheorem 3 Let {QJbe PPSNs aboutPFreely select mi distinct algebraic hyperplanes 1, thenmust be a PPSN about Pmi,...