The Study Of Several Problems On Ideal Interprolation

The definition of ideal interpolation was put forward to unify the study of polynomial interpolation. The problem of polynomial interpolation is ideal interpolation if and only if its interpolation conditions consist of a family of in-terpolation points, and for each point, a family of derivative conditions induced by a finite-dimensional D-invariant polynomial subspace. Ideal interpolation is defined by an ideal projector. An ideal projector is a linear idempotent operator on the polynomial space whose kernel is an ideal. Theoretical study of ideal interpolation mainly includes two aspects:Discussing the relation be-tween ideal projectors and Hermite projectors; Investigating the error formu-las for ideal projectors. In the actual application, given a set of interpolation conditions satisfying ideal interpolation, one can determine various forms of interpolation bases. The method of computing interpolation bases can be not only exact, but also approximate. The computation of approximate interpo-lation bases involves the computation of approximate vanishing ideals. With the aid of algebraic geometry and numerical computation, we discuss several problems on the study of ideal interpolation. The main results of this paper are as follows:Chapter 2 analyzes the point sets with unique associated monomial quo-tient bases. The point sets with unique associated monomial quotient bases belong to a special class of point sets, whose vanishing ideals have unique monomial quotient bases. Using algebraic geometry, we give a useful criterion for a zero-dimensional ideal with a unique monomial quotient basis. Let-?i be an elimination order for xi, then a zero-dimensional ideal I C F[x] has a unique monomial quotient basis if and only if the Grobner escaliers of I w.r.t. -