The Discretization for a Special Class of Ideal Projectors

Polynomial interpolation is to construct a polynomial p belonging to a finite-dimensional subspace of F x from a set of data that agrees with a given function f at the data set, where F x : F x1, . . . , xd denotes the polynomial ring in d variables over the field F. It is important to make the comment that F is the real field R or the complex field C 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 F x , namely, a linear idempotent operator on F x whose kernel is an ideal, cf. 3 . When the kernel of an ideal projector P is the vanishing ideal of certain finite set Ξ in F, P is a Lagrange projector which provides the Lagrange interpolation on Ξ. 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 P on F x , the kernel of P is ideal if and only if the range of its dual projector is of the form:


Introduction
Polynomial interpolation is to construct a polynomial p belonging to a finite-dimensional subspace of F x from a set of data that agrees with a given function f at the data set, where F x : F x 1 , . . ., x d denotes the polynomial ring in d variables over the field F. It is important to make the comment that F is the real field R or the complex field C 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 F x , namely, a linear idempotent operator on F x whose kernel is an ideal, cf. 3 .When the kernel of an ideal projector P is the vanishing ideal of certain finite set Ξ in F d , P is a Lagrange projector which provides the Lagrange interpolation on Ξ.
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 P on F x , the kernel of P is ideal if and only if the range of its dual projector is of the form: 1.1 with some finite point set Ξ ⊂ F d , D-invariant finite-dimensional polynomial subspace Q ξ ⊂ F x for each ξ ∈ Ξ. δ ξ denotes the evaluation functional at the point ξ, and Q ξ D will be explained in the next section.
In the univariate case, for an integer n, there is only one D-invariant polynomial subspace of degree less than n, 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 C x is an Hermite projector in 9 .
However, Shekhtman 10 constructed a counterexample to this conjecture for every d ≥ 3.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 n 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 D-invariant polynomial subspaces.Sections 3 and 4 discuss these two classes of D-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.

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 N 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, α α 1 , . . ., α d .For arbitrary α ∈ N d , we define α! α Henceforward, ≤ will denote the usual product order on N d , that is, for arbitrary α, A monomial x α ∈ F x is a power product of the form x α for its associated differential operator.For a finite-dimensional polynomial subspace Q, we define If the polynomial subspace Q is closed under differentiation, then we call it a D-invariant polynomial subspace.
Let P be a finite-rank ideal projector on F x .The range and kernel of P will be described as ran P : g ∈ F x : g Pf for some f ∈ F x , ker P : g ∈ F x : Pg 0 .

2.4
Let P be the dual projector of P , then ran P is the set of interpolation conditions matched by P .If q q 1 , . . ., q s is an F-basis for ran P and λ λ 1 , . . ., λ s an F-basis for ran P , then their Gram matrix: λ T q : λ i q j 1≤i,j≤s 2.5 is invertible.2.6

as above, and let
where • denotes the Euclidean product.Then p D D α ρ .
Given a lower set d ⊂ N d and ρ ρ 1 , ρ 2 , . . ., ρ d ∈ F d d as above, then the subspace: The following proposition discretizes this functional subspace δ ξ Q D into a sequence of evaluation functional subspaces. where

and the remainder O h is a polynomial in h.
Proof.For convenience, let Then we can obtain that for m 0, . . ., α 1 : In the following, we will simplify the right-hand side of the above equality.There are two cases which must be examined.
In this case, there must exist some 1 ≤ j ≤ d such that d i 1 γ i,j < α j .By Lemma 2.1, it follows for such j: 3.8 Thus, for all 0 ≤ m ≤ α 1 − 1, d m ϕ 0 /dh m 0.
Case 2 m α 1 .In this case, if there exists some 1 ≤ j ≤ d such that d i 1 γ i,j > α j , then there must exist another 1 ≤ j ≤ d, such that d i 1 γ i,j < α j .Hence, when m α 1 , if and only if for all 1 ≤ j ≤ d, d i 1 γ i,j α j .Combining this with the fact: we can conclude that

3.11
That is In sum, we have deduced that 3.14

The Second Class of D-Invariant Polynomial Subspaces
Let a a 0 , a 1 , . . ., a n with n ≥ 1 be an n 1-tuple of positive integers satisfying and the map τ be as above.Let q n,m , m 0, 1, . . ., a 1 be polynomials defined by i,j , and let Q be a polynomial subspace defined by

4.5
Then, the following hold: ii Q is a D-invariant polynomial subspace.
Proof.i We assume that there exist k Since there exists some 1 ≤ i 0 ≤ d such that c i 0 ,0 / 0, then x m i 0 must belong to the support of q n,m .Using this together with the definition of τ, we conclude that the degree of q n,m is m.Specifically, x a 1 i 0 belongs to the support of q n,a 1 , while for all 0 ≤ m ≤ a 1 − 1, x a 1 i 0 cannot belong to the support of q n,m .Therefore k a 1 0, which implies that Arguing for m a 1 − 1, a 1 − 2, . . ., 1, 0 as for m a 1 , we get k a 1 −1 0, k a 1 −2 0, . . ., k 1 0, k 0 0, successively.That is to say, the family of polynomials q n,m , 0 ≤ m ≤ a 1 is F-linearly independent.So dim Q a 1 1.

4.8
Consequently, Next, let m be an arbitrary integer satisfying a n ≤ m ≤ a 1 , and s the minimum integer between 1 and n such that m − a s ≥ 0. We claim

4.10
This claim together with equality 4.9 immediately means that Q is D-invariant.
To prove our claim, we will use induction on the number n.When n 1, our claim can be easily verified.Now, assume that our claim is true for n − 1.To prove that it holds for n, let k be the maximum nonnegative integer such that ka n ≤ m, that is, and k the maximum nonnegative integer such that k a n ≤ m − 1.From the definition of q n,m , we obtain that

4.12
That is which plays an important role in what follows.At this point, we have two cases to consider.
Case 1 a n ≤ m < a n−1 .In this case, s n and 0 ≤ m − la n < a n−1 for all 0 ≤ l ≤ k.Hence, By 4.13 , we get So we obtain that

4.16
Case 2 a n−1 ≤ m ≤ a 1 .In this case, 0 ≤ s ≤ n − 1.By 4.11 and 4.1 , it follows that m−ka n −a n−1 < 0. Thus, there exists some 0 As a result, we have that 0 ≤ m − la n < a n−1 for all l 0 1 ≤ l ≤ k, which implies that Therefore,

4.19
For all 0 ≤ l ≤ l 0 , we have a n−1 ≤ m − la n ≤ a 1 .Then our inductive hypothesis implies that where s l is the minimal integer between 1 and n − 1 such that m − la n − a s l ≥ 0. It should be noticed that s 0 s.

4.21
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 s 0 ≤ j ≤ n−1, let k j denote the maximum nonnegative integer satisfying k j a n ≤ m − a j , that is,

4.22
From 4.17 , we know k n−1 l 0 .Due to 4.13 , we observe that

4.24
Here, we have used the fact that q n−1,m−a j −la n 0 for all s 0 ≤ j ≤ n − 1, k j 1 ≤ l ≤ l 0 .Now, recall that s l , 0 ≤ l ≤ l 0 is the minimal integer between 1 and n − 1 such that m − la n − a s l ≥ 0, then 4.25

ISRN Applied Mathematics
According to 4.24 , 4.25 , and the fact s s 0 , we find that

4.27
Proposition 4.2.Let a a 0 , a 1 , . . ., a n , c i c i,0 , c i,1 , . . ., c i,n with 1 ≤ i ≤ d, and let q n,m with 0 ≤ m ≤ a 1 be as above, h a nonzero number in F. Then for arbitrary p ∈ F x and ξ ∈ F d :

4.28
Proof.Applying Taylor Formulas, we obtain where c By means of the map τ defined as in 4.3 , the above equality can be rewritten as

4.30
where O h m 1 is a polynomial in h.
and let P h be a Lagrange projector with

5.3
Then, the following statements hold: i there exists a positive η ∈ F such that ran P h ran P, ∀ 0 < |h| < η; 5.4 ii P is the pointwise limit of P h , 0 < |h| < η, as h tends to zero.
Proof.Firstly, one can easily verify that form F-bases for ran P and ran P h , respectively, where Provided that the entries of λ and λ h are arranged in the same order, namely, for arbitrary fixed 1 ≤ k ≤ μ, α ∈ d k , the corresponding entries of λ and λ h are in the same position, the same as for arbitrary fixed 1 ≤ l ≤ ν, 0 ≤ m ≤ a l 1 .We denote λ λ 1 , . . ., λ s and λ h λ h,1 , . . ., λ h,s , respectively.Let q q 1 , q 2 , . . ., q s be an F-basis for ranP , For convenience, we introduce two s × s matrices: λ T q : λ i q j 1≤i,j≤s , λ T h q : λ h,i q j 1≤i,j≤s 5.7 and for arbitrary f ∈ F x , vectors: For arbitrary p ∈ F x , we have the following facts according to equality 3.5 and 4.28 .
1 For fixed 1 ≤ k ≤ μ and α ∈ d k , D α ρ k p ξ k can be linearly expressed by since d k is lower, and moreover, the linear combination coefficient of each 2 For fixed 1 ≤ l ≤ ν and 0 ≤ m ≤ a l 1 , q l n l ,m D p ξ μ l can be linearly expressed by p ξ μ l φ l rh : 0 ≤ r ≤ m ∪ {O h }.

5.10
Also, the linear combination coefficient of each p ξ μ l φ l rh is independent of p ∈ F x .
In brief, we can conclude that there exists a nonsingular matrix T such that where each entry of E h | h has the same order as h.As a consequence, the linear systems: 12 are equivalent, namely, they have the same set of solutions.
i From 5.11 , it follows that each entry of matrix λ T h q converges to its corresponding entry of matrix λ T q as h tends to zero, which implies that lim h → 0 det λ T h q det λ T q .5.13 Since det λ T q / 0, there exists η > 0 such that det λ T h q / 0, 0 < |h| < η.

5.14
Notice that 5.11 directly leads to rank λ T h q rank λ T h q : ran P h span F q, 0 < |h| < η, 5.15 follows, that is, q forms an F-basis for ran P h .Since q is also an F-basis for ran P , we have ran P ran P h , 0 < |h| < η.

5.16
ii Suppose that x and x 0 be the unique solutions of nonsingular linear systems: λ T h q x λ T h f, 5.17 λ T q x λ T f, 5.18 respectively, where f ∈ F x and 0 < |h| < η.It is easy to see that P h f q x, Pf qx 0 .

ISRN Applied Mathematics
Notice that, as h → 0, P is the pointwise limit of P h if and only if Pf is the coefficient-wise limit of P h f for all f ∈ F x .Therefore, it is sufficient to show that for every f ∈ F x , the solution vector of system 5.17 converges to the one of system 5.18 as h tends to zero, namely, lim h → 0 x x 0 .5.20 By 5.11 , the linear system: λ T h q x λ T h f 5.21 can be rewritten as λ T q E h x λ T f h .

5.22
Since system 5.21 is equivalent to system 5.17 , x 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

5.23
Since each component of vector h − E h x 0 has the same order as h, it follows that lim h → 0 x − x 0 0, or, equivalently, lim h → 0 x x 0 , which completes the proof of the theorem.

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

Lemma 2 . 1
see 15, page 90 .Let n and m be arbitrary nonnegative integers satisfying n ≤ m.

By 4 .
19 and 4.20 , we can deduce that

Theorem 5 . 1 .
For each 1 ≤ k ≤ μ, assume that d k ⊂ N d is a lower set, and ρ k ρ k 1 , . . ., ρ k d ∈ F d d satisfies the conditions of Section 3. Likewise, for each 1 ≤ l ≤ ν, assume that a , . . ., c l i,n l , 1 ≤ i ≤ d, and q l n l ,m , 0 ≤ m ≤ a l 1 satisfy the conditions of Section 4. With the notation above, let P be an ideal projector with