ISRN Applied Mathematics

Volume 2012 (2012), Article ID 359069, 15 pages

doi:10.5402/2012/359069

Research Article

The Discretization for a Special Class of Ideal Projectors

Zhe Li,¹ Shugong Zhang,² and Tian Dong²

¹Department of Applied Mathematics, School of Science, Changchun University of Science and Technology, Changchun 130022, China

²Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China

Received 21 March 2012; Accepted 9 April 2012

Academic Editors: J. Kou, M. Qatu, and R. Samtaney

Copyright © 2012 Zhe Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We focus on a special class of ideal projectors, subspaces, which possesses two classes of D-invariant polynomial subspaces. The first is a classical type, while the second is a new class. With matrix computation, we discretize this class of ideal projectors into a sequence of Lagrange projectors.

1. Introduction

Polynomial interpolation is to construct a polynomial belonging to a finite-dimensional subspace of from a set of data that agrees with a given function at the data set, where denotes the polynomial ring in variables over the field . It is important to make the comment that is the real field or the complex field in this paper. Univariate polynomial interpolation has a well-developed theory, while the multivariate one is very problematic since a multivariate interpolation polynomial is determined not only by the cardinal but also by the geometry of the data set, cf. [1, 2].

Recently, more and more people are getting interested in ideal interpolation, which is defined by an ideal projector on , namely, a linear idempotent operator on whose kernel is an ideal, cf. [3]. When the kernel of an ideal projector is the vanishing ideal of certain finite set in , is a Lagrange projector which provides the Lagrange interpolation on .

It is well known that an ideal projector can be characterized completely by the range of its dual projector, cf. [4–7]. Given a finite-rank linear projector on , the kernel of is ideal if and only if the range of its dual projector is of the form:

with some finite point set , -invariant finite-dimensional polynomial subspace for each . denotes the evaluation functional at the point , and will be explained in the next section.

In the univariate case, for an integer , there is only one -invariant polynomial subspace of degree less than , which implies that every univariate ideal projector can be viewed as a limiting case of Lagrange projectors, cf. [8]. This prompted de Boor to define the Hermite projector as the pointwise limit of Lagrange projectors and conjecture that every ideal projector in is an Hermite projector in [9].

However, Shekhtman [10] constructed a counterexample to this conjecture for every . In the same paper, Shekthman also showed that the conjecture is true for bivariate complex projectors with the help of Fogarty Theorem (see [11]). Later, using linear algebra, de Boor and Shekhtman [12] reproved the same result. Specifically, Shekhtman [13] completely analyzed the bivariate ideal projectors which are onto the space of polynomials of degree less than over real or complex field and verified the conjecture in this particular case.

In this paper, we consider the converse problem: how to generate a sequence of Lagrange projectors practically such that this sequence of Lagrange projectors converges pointwise to a given Hermite projector. We focus on a special class of ideal projectors, which possesses two classes of -invariant polynomial subspaces. Sections 3 and 4 discuss these two classes of -invariant subspaces, respectively. In Section 5, we discretize this class of ideal projectors into a sequence of Lagrange projectors. The next section is devoted a preparation for the paper.

2. Preliminaries

In this section, we will introduce some notation used throughout the paper and give some background results. For more details, we refer the reader to [9, 14].

We use to stand for the monoid of nonnegative integers and boldface type for tuples with their entries denoted by the same letter with subscripts, for example, . For arbitrary , we define .

Henceforward, ≤ will denote the usual product order on , that is, for arbitrary , , if and only if . A finite subset is lower if for every , implies .

A monomial is a power product of the form with . Thus, a polynomial in can be expressed as

with nonzero. For a polynomial as in (2.1), we write

for its associated differential operator. For a finite-dimensional polynomial subspace , we define

If the polynomial subspace is closed under differentiation, then we call it a -invariant polynomial subspace.

Let be a finite-rank ideal projector on . The range and kernel of will be described as

Let be the dual projector of , then is the set of interpolation conditions matched by . If is an -basis for and an -basis for , then their Gram matrix:

is invertible.

Lemma 2.1 (see [15, page 90]). Let and be arbitrary nonnegative integers satisfying . Then

3. The First Class of -Invariant Polynomial Subspaces

Let and in which

are -linearly independent unit vectors. Then we define a differential operator by the formula:

Lemma 3.1. Let , be as above, and let

where denotes the Euclidean product. Then .

Given a lower set and as above, then the subspace:

is obviously -invariant. The following proposition discretizes this functional subspace into a sequence of evaluation functional subspaces.

Proposition 3.2. Let be as above, a nonzero number in , and let , . Then for arbitrary

where , and the remainder is a polynomial in .

Proof. For convenience, let

Then we can obtain that for :

with , .

In the following, we will simplify the right-hand side of the above equality. There are two cases which must be examined.

Case 1 (). In this case, there must exist some such that . By Lemma 2.1, it follows for such :

Thus, for all , .

Case 2 (). In this case, if there exists some such that , then there must exist another , such that . Hence, when ,

if and only if for all , . Combining this with the fact:

we can conclude that

That is

In sum, we have deduced that

which implies that

4. The Second Class of -Invariant Polynomial Subspaces

Let with be an -tuple of positive integers satisfying

and let

satisfying that are not all zero. Then it is clear that

with defines a map .

Proposition 4.1. Let , , , and the map be as above. Let , be polynomials defined by

with , , and let be a polynomial subspace defined by

Then, the following hold:

(i);(ii) is a -invariant polynomial subspace.

Proof. (i) We assume that there exist such that

Since there exists some such that , then must belong to the support of . Using this together with the definition of , we conclude that the degree of is .

Specifically, belongs to the support of , while for all , cannot belong to the support of . Therefore , which implies that

Arguing for as for , we get , successively. That is to say, the family of polynomials is -linearly independent. So .

(ii) If , then implies that with . Thus, for all :

Consequently,

Next, let be an arbitrary integer satisfying , and the minimum integer between 1 and such that . We claim

This claim together with equality (4.9) immediately means that is -invariant.

To prove our claim, we will use induction on the number . When , our claim can be easily verified. Now, assume that our claim is true for . To prove that it holds for , let be the maximum nonnegative integer such that , that is,

and the maximum nonnegative integer such that . From the definition of , we obtain that

That is

which plays an important role in what follows. At this point, we have two cases to consider.

Case 1 (). In this case, and for all . Hence,

By (4.13), we get

So we obtain that

Case 2 (). In this case, .

By (4.11) and (4.1), it follows that . Thus, there exists some such that

As a result, we have that for all , which implies that

Therefore,

For all , we have . Then our inductive hypothesis implies that

where is the minimal integer between 1 and such that . It should be noticed that .

By (4.19) and (4.20), we can deduce that

It remains to show that the last row of the above equality and the right-side hand of (4.10) are equal. More precisely, for each , let denote the maximum nonnegative integer satisfying , that is,

From (4.17), we know .

Due to (4.13), we observe that

Furthermore,

Here, we have used the fact that for all , .

Now, recall that is the minimal integer between 1 and such that , then

According to (4.24), (4.25), and the fact , we find that

Consequently,

Proposition 4.2. Let , with , and let with be as above, a nonzero number in . Then for arbitrary and :

Proof. Applying Taylor Formulas, we obtain

where , .

By means of the map defined as in (4.3), the above equality can be rewritten as

where is a polynomial in .

Finally, by Lemma 2.1, we can conclude that

5. Main Theorem

Let be distinct points. For each , assume that is a lower set, and satisfies the conditions of Section 3. Likewise, for each , assume that , , , , and , satisfy the conditions of Section 4.

Theorem 5.1. With the notation above, let be an ideal projector with

and let be a Lagrange projector with

where and

Then, the following statements hold:

(i)there exists a positive such that

(ii) is the pointwise limit of , as tends to zero.

Proof. Firstly, one can easily verify that

form -bases for and , respectively, where

Provided that the entries of and are arranged in the same order, namely, for arbitrary fixed , , the corresponding entries of and are in the same position, the same as for arbitrary fixed , . We denote and , respectively.

Let be an -basis for , For convenience, we introduce two matrices:

and for arbitrary , vectors:

For arbitrary , we have the following facts according to equality (3.5) and (4.28).

(1)For fixed and , can be linearly expressed by

since is lower, and moreover, the linear combination coefficient of each is independent of .(2) For fixed and , can be linearly expressed by

Also, the linear combination coefficient of each is independent of .

In brief, we can conclude that there exists a nonsingular matrix such that

where each entry of has the same order as . As a consequence, the linear systems:

are equivalent, namely, they have the same set of solutions.

(i)From (5.11), it follows that each entry of matrix converges to its corresponding entry of matrix as tends to zero, which implies that

Since , there exists such that

Notice that (5.11) directly leads to :

follows, that is, forms an -basis for . Since is also an -basis for , we have

(ii) Suppose that and be the unique solutions of nonsingular linear systems:

respectively, where and . It is easy to see that

Notice that, as , is the pointwise limit of if and only if is the coefficient-wise limit of for all . Therefore, it is sufficient to show that for every , the solution vector of system (5.17) converges to the one of system (5.18) as tends to zero, namely,

By (5.11), the linear system:

can be rewritten as

Since system (5.21) is equivalent to system (5.17), is also the unique solution of it. Consequently, using the perturbation analysis of the sensitivity of linear systems (see, e.g., [16, page 80]), we have

Since each component of vector has the same order as , it follows that , or, equivalently, , which completes the proof of the theorem.

6. Conclusions

The main work of this paper is to discretize a class of ideal projectors with two special classes of -invariant polynomial subspaces into a sequence of Lagrange projectors.

Acknowledgment

This work was supported by the National Natural Science Foundation of China nos. 11101185 and 11171133.

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