The Study Of Error Formulas And Discretization Problem In Ideal Interpolation

For studying a common multivariate polynomial interpolation problem, Birkhoff first introduced the definition of ideal interpolation. Ideal interpo-lation is a linear interpolation scheme whose interpolated functions are poly-nomials. Ideal interpolation can be seen as the Lagrange interpolation and Hermite interpolation in the multivariate case. Specifically, ideal interpolation is defined by an ideal projector. An ideal projector is a linear idempotent pro-jector on the polynomial space whose kernel is an ideal. In the theory of ideal interpolation, the interpolation space is the range of the ideal projector, and the functional space of interpolation conditions is the range of the dual of the ideal projector. The functional space of interpolation conditions consists of a set of interpolation sites, and for each site, an evaluation functional and some differential operators induced by a finite dimensional D-invariant subspace. A .D-invariant subspace is a linear space of polynomials and it is closed under dif-ferentiation. Since the concept of "D-invariant" is a generalized understanding of the "consecutive order derivatives" in the interpolation conditions of univari-ate Hermite interpolation, ideal interpolation includes the classical Lagrange interpolation and Hermite interpolation. An ideal projector corresponding to a Lagrange interpolation problem is called Lagrange projector.In 2005, de Boor raised the following problems in his overview on ide-al interpolation, the first one:Do ideal projectors have a uniform structure expression of error formulas? The second one:Which ideal projectors have "good" error formulas? The third one:If an ideal projector is Hermite pro-jector, how to find Lagrange projectors that converge to it. Until now, these problems are still the research focus in ideal interpolation. For convenience, we call the fist two problems the error formulas problem and the last one the discretization problem. In this thesis, we study these problems with the aid of algebraic geometry. The main results are as follows:1. We present a uniform structure expression of error formulas for ide-al projectors. As we know, the structure expression of error formulas for univariate ideal projectors is simple. In order to extend it to the multivari-ate case, de Boor presented the definition of "good" error formulas. The "good" error formulas are error structure expressions. Specifically, it refers to there exist homogeneous polynomials Hj and linear operators Cj such that f-Pf=?j=1m Cj(Hj(D)f)hj and Hj(D)hk=?j,k, where f is an interpolat-ed function, P is an ideal projector, Hj(D) is the differential operator, and {h1,...,hm} is an ideal basis for kerP. de Boor expected that such formulas hold for all ideal projectors. However, Shekhtman provided a bivariate coun-terexample, and believes that most ideal projectors do not have the "good" error formulas. We study the algebraic structure of error formulas for ideal projectors. On the basis of "good" error formulas, we propose the concept of "normal" error formulas. Using reduction theory, we prove that the lexico-graphic order reduced Grobner basis admits such a "normal" error formula for all ideal interpolation. Using univariate B-spline, we discuss the Shekhtman's example and give an explicit form of "normal" error formula for this example.2. We present the explicit forms of "good" error formulas for a class of ideal projectors. As we mentioned earlier, not all the ideal projectors have the "good" error formulas. Till now, the research of "good" error formulas has made some progress. Shekhtman showed that the "good" error formulas hold for a special class of interpolation sites. Li proved that there exists a "good" error formula if the kernel of an ideal projector has a universal Grobner basis. de Boor gave the explicit forms of "good" error formulas for tensor-product and Chung-Yao interpolation. Inspired by these works, we study the "good" error formulas for a class of ideal projectors. For Lagrange interpolation determined by Cartesian sets, we get the remainder of interpolation by divided differences algorithm. Then we transform the remainder of interpolation into a "good" error formula. Using the relationship between the divided differences and the B-spline, we present the explicit form of this "good" error formula.3. We discuss the discretization problem for a class of bivariate Hermite projectors. When extending a concept, people always retain the original struc-ture properties. In the univariate case, Hermite interpolation can be viewed as the pointwise limit of Lagrange interpolation. Thus de Boor defines Her-mite projector as the pointwise limit of Lagrange projectors. While all the univariate ideal projectors are Hermite projectors, and it is still true for some multivariate cases, there are examples to show that multivariate non Hermite projectors do exist. So one is interested in studying what ideal projectors are Hermite projectors and how to find Lagrange projectors that converge to it. Around.this problem (the discratization problem), de Boor and Shekhtman proved that all the bivariate ideal projectors are Hermite projectors. Also they propose an algorithm for computing Lagrange projectors that converge to Hermite projector. However, the algorithm is very complicated and is not easy to implement. We study a class of bivariate Hermite projectors with the interpolation conditions ?? ?(D), ?(n):= F<n[x,y] (?) spanF{pn}, where ?? is the evaluation functional at ?, D is the differentiation symbol, F<n[x,y] denotes the set of all the bivariate polynomials of degree less than n, and pn is an arbitrary bivariate polynomial of degree n. For a given Hermite projector as above, we propose a simple and effective algorithm for computing Lagrange projectors that converge to it.