# The Discretization for a Special Class of Ideal Projectors

Authors: Zhe Li; Shugong Zhang; Tian Dong
Journal: ISRN Applied Mathematics
 (2012)

---

## Abstract

We focus on a special class of ideal projectors, subspaces, which possesses two classes ofD-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.


---

## Body

## 1. Introduction

Polynomial interpolation is to construct a polynomialp 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 anideal 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 𝔽d, 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)⨁ξ∈ΞδξQξ(D),
with some finite point set Ξ⊂𝔽d, D-invariant finite-dimensional polynomial subspace 𝒬ξ⊂𝔽[x] for each ξ∈Ξ. δξ denotes the evaluation functional at the point ξ, and 𝒬ξ(D) will be explained in the next section.In the univariate case, for an integern, 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 ofD-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.


## 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 useℕ 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=(α1,…,αd). For arbitrary α∈ℕd, we define α!=α1!⋯αd!.Henceforward, ≤ will denote the usual product order onℕd, that is, for arbitrary α, β∈ℕd, α≤β if and only if αi≤βi,i=1,…,d. A finite subset 𝒜⊂ℕd is lower if for every α∈𝒜, 0≤β≤α implies β∈𝒜.A monomialxα∈𝔽[x] is a power product of the form x1α1⋯xdαd with α∈ℕd. Thus, a polynomial p in 𝔽[x] can be expressed as(2.1)p=∑α∈Ndp̂(α)xα,
with p̂(α)∈𝔽 nonzero. For a polynomial p(x) as in (2.1), we write(2.2)p(D):F[x]⟶F[x]f⟼∑α∈Ndp̂(α)∂∥α∥1f∂x1α1⋯∂xdαd,
for its associated differential operator. For a finite-dimensional polynomial subspace 𝒬, we define(2.3)Q(D):=spanF{q(D):q∈Q}.
If the polynomial subspace 𝒬 is closed under differentiation, then we call it a D-invariant polynomial subspace.LetP be a finite-rank ideal projector on 𝔽[x]. The range and kernel of P will be described as(2.4)ranP:={g∈F[x]:g=Pfforsomef∈F[x]},ker⁡P:={g∈F[x]:Pg=0}.
Let P′ be the dual projector of P, then ranP′ is the set of interpolation conditions matched by P. If q=(q1,…,qs) is an 𝔽-basis for ranP and λ=(λ1,…,λs) an 𝔽-basis for ranP′, then their Gram matrix:
(2.5)λTq:=(λiqj)1≤i,j≤s
is invertible.Lemma 2.1 (see [15, page 90]).
Letn and m be arbitrary nonnegative integers satisfying n≤m. Then
(2.6)∑r=0m(-1)m-r(mr)rn={m!,n=m;0,0≤n<m.


## 3. The First Class ofD-Invariant Polynomial Subspaces

Letα=(α1,…,αd)∈ℕd and ρ=(ρ1,ρ2,…,ρd)∈(𝔽d)d in which(3.1)ρ1=(ρ1,1,…,ρ1,d),ρ2=(ρ2,1,…,ρ2,d),…,ρd=(ρd,1,…,ρd,d)
are 𝔽-linearly independent unit vectors. Then we define a differential operator Dρα:𝔽[x]→𝔽[x] by the formula:
(3.2)Dραf=∂‖α‖1f∂ρ1α1⋯∂ρdαd.Lemma 3.1.
Letα=(α1,…,αd)∈ℕd, ρ=(ρ1,ρ2,…,ρd)∈(𝔽d)d be as above, and let
(3.3)p=∏i=1d(ρi⋅x)αi,
where · denotes the Euclidean product. Then p(D)=Dρα.Given a lower set𝔡⊂ℕd and ρ=(ρ1,ρ2,…,ρd)∈(𝔽d)d as above, then the subspace:(3.4)Q=spanF{∏i=1d(ρi⋅x)αi:α∈d}
is obviously D-invariant. The following proposition discretizes this functional subspace δξ𝒬(D) into a sequence of evaluation functional subspaces.Proposition 3.2.
Letρ=(ρ1,ρ2,…,ρd)∈(𝔽d)d be as above, h a nonzero number in 𝔽, and let α∈ℕd, ξ∈𝔽d. Then for arbitrary p∈𝔽[x](3.5)1h∥α∥1∑0≤β≤α(-1)‖α-β‖1(αβ)p(ξ+h∑i=1dβiρi)=(Dραp)(ξ)+O(h),
where (αβ)=∏i=1d(αiβi), and the remainder O(h) is a polynomial in h.Proof.
For convenience, let(3.6)φ=∑0≤β≤α(-1)‖α-β‖1(αβ)p(ξ+h∑k=1dβkρk).
Then we can obtain that for m=0,…,∥α∥1:
(3.7)dmφ(0)dhm=∑‖γ1‖1+⋯+‖γd‖1=mρ1,1γ1,1⋯ρd,1γ1,d⋯ρ1,dγd,1⋯ρd,dγd,d∂mp(ξ)∂x1‖γ1‖1…∂xd‖γd‖1×(mγ1,1⋯γ1,d,…,γd,1,…,γd,d)∏j=1d(∑βj=0αj(-1)αj-βj(αjβj)βj∑i=1dγi,j),
with γi=(γi,1,…,γi,d)∈ℕd, 1≤i≤d.
In the following, we will simplify the right-hand side of the above equality. There are two cases which must be examined.
Case 1 (0≤m≤∥α∥1-1).
In this case, there must exist some1≤j≤d such that ∑i=1dγi,j<αj. By Lemma 2.1, it follows for such j:
(3.8)∑βj=0αj(-1)αj-βj(αjβj)βj∑i=1dγi,j=0.
Thus, for all 0≤m≤∥α∥1-1, dmφ(0)/dhm=0.Case 2 (m=∥α∥1).
In this case, if there exists some1≤j≤d such that ∑i=1dγi,j>αj, then there must exist another 1≤j′≤d, such that ∑i=1dγi,j′<αj′. Hence, when m=∥α∥1,
(3.9)∏j=1d(∑βj=0αj(-1)αj-βj(αjβj)βj∑i=1dγi,j)≠0
if and only if for all 1≤j≤d, ∑i=1dγi,j=αj. Combining this with the fact:
(3.10)∑βj=0αj(-1)αj-βj(αjβj)βjαj=αj!,1≤j≤d,
we can conclude that
(3.11)d‖α‖1φ(0)dh‖α‖1=‖α‖1!∑γ1,1+⋯+γd,1=α1⋯∑γ1,d+⋯+γd,d=αdρ1,1γ1,1⋯ρd,1γ1,d⋯ρ1,dγd,1⋯ρd,dγd,d×(α1γ1,1,…,γd,1)⋯(αdγ1,d,…,γd,d)∂∥α∥1p(ξ)∂x1‖γ1‖1⋯∂xd‖γd‖1.
That is
(3.12)d‖α‖1φ(0)dh‖α‖1=‖α‖1!(Dραp)(ξ).
In sum, we have deduced that
(3.13)dmφ(0)dhm={0,0≤m≤‖α‖1-1;‖α‖1!(Dραp)(ξ),m=‖α‖1,
which implies that
(3.14)φ=h‖α‖1(Dραp)(ξ)+O(h‖α‖1+1).


## 4. The Second Class ofD-Invariant Polynomial Subspaces

Leta=(a0,a1,…,an) with n≥1 be an n+1-tuple of positive integers satisfying(4.1)a0=1,a1>⋯>an≥2,
and let
(4.2)c1=(c1,0,c1,1,…,c1,n),…,cd=(cd,0,cd,1,…,cd,n)∈Fn+1
satisfying that c1,0,c2,0,…,cd,0 are not all zero. Then it is clear that(4.3)τ(γ1,…,γd)=∑j=0naj∑i=1dγi,j,
with γi=(γi,0,γi,1,…,γi,n)∈ℕn+1 defines a map τ:(ℕn+1)d→ℕ.Proposition 4.1.
Leta=(a0,a1,…,an), ci=(ci,0,ci,1,…,ci,n), 1≤i≤d, and the map τ be as above. Let qn,m, m=0,1,…,a1 be polynomials defined by
(4.4)qn,m=∑τ(γ1,…,γd)=mc1γ1⋯cdγdγ1!⋯γd!x1‖γ1‖1⋯xd‖γd‖1,
with γi=(γi,0,γi,1,…,γi,n)∈ℕn+1, ciγi=∏j=0nci,jγi,j, and let 𝒬 be a polynomial subspace defined by
(4.5)Q=spanF{qn,m:0≤m≤a1}.
Then, the following hold:(i)
dim𝒬=a1+1;(ii)
𝒬 is a D-invariant polynomial subspace.Proof.
(i) We assume that there existk0,k1,…,ka1-1,ka1∈𝔽 such that
(4.6)k0qn,0+k1qn,1+⋯+ka1-1qn,a1-1+ka1qn,a1=0.
Since there exists some 1≤i0≤d such that ci0,0≠0, then xi0m must belong to the support of qn,m. Using this together with the definition of τ, we conclude that the degree of qn,m is m.
Specifically,xi0a1 belongs to the support of qn,a1, while for all 0≤m≤a1-1, xi0a1 cannot belong to the support of qn,m. Therefore ka1=0, which implies that
(4.7)k0qn,0+k1qn,1+⋯+ka1-1qn,a1-1=0.
Arguing form=a1-1,a1-2,…,1,0 as for m=a1, we get ka1-1=0,ka1-2=0,…,k1=0,k0=0, successively. That is to say, the family of polynomials qn,m,0≤m≤a1 is 𝔽-linearly independent. So dim𝒬=a1+1.
(ii) If1≤m<an, then τ(γ1,…,γd)=m implies that γi,j=0 with 1≤i≤d,1≤j≤n. Thus, for all 1≤m<an:
(4.8)qn,m=∑γ1,0+⋯+γd,0=mc1,0γ1,0⋯cd,0γd,0γ1,0!⋯γd,0!x1γ1,0⋯xdγd,0=1m!(c1,0x1+c2,0x2+⋯+cd,0xd)m.
Consequently,
(4.9)∂qn,m∂xi=ci,0qn,m-1,1≤m<an.
Next, letm be an arbitrary integer satisfying an≤m≤a1, and s the minimum integer between 1 and n such that m-as≥0. We claim
(4.10)∂qn,m∂xi=ci,0qn,m-1+∑j=snci,jqn,m-aj,an≤m≤a1.
This claim together with equality (4.9) immediately means that 𝒬 is D-invariant.
To prove our claim, we will use induction on the numbern. 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 kan≤m, that is,
(4.11)kan≤m,(k+1)an>m
and k′ the maximum nonnegative integer such that k′an≤m-1. From the definition of qn,m, we obtain that
(4.12)qn,m=∑l=0k(∑τ((γ1,0,…,γ1,n-1),…,(γd,0,…,γd,n-1))=m-lanc1,0γ1,0⋯c1,n-1γ1,n-1⋯cd,0γd,0⋯cd,n-1γd,n-1γ1,0!⋯γ1,n-1!⋯γd,0!⋯γd,n-1!×x1γ1,0+⋯+γ1,n-1⋯xdγd,0+⋯+γd,n-1∑τ((γ1,0,…,γ1,n-1),…,(γd,0,…,γd,n-1))=m-lanc1,0γ1,0⋯c1,n-1γ1,n-1⋯cd,0γd,0⋯cd,n-1γd,n-1γ1,0!⋯γ1,n-1!⋯γd,0!⋯γd,n-1!)(∑γ1,n+⋯+γd,n=lc1,nγ1,n⋯cd,nγd,nc1,n!⋯cd,n!x1γ1,n⋯xdγd,n).
That is
(4.13)qn,m=∑l=0k1l!(c1,nx1+⋯+cd,nxd)lqn-1,m-lan,
which plays an important role in what follows. At this point, we have two cases to consider.
Case 1 (an≤m<an-1).
In this case,s=n and 0≤m-lan<an-1 for all 0≤l≤k. Hence,
(4.14)qn-1,m-lan=1(m-lan)!(c1,0x1+⋯+cd,0xd)m-lan,0≤l≤k.
By (4.13), we get
(4.15)qn,m=∑l=0k1l!(m-lan)!(c1,nx1+⋯+cd,nxd)l(c1,0x1+⋯+cd,0xd)m-lan.
So we obtain that
(4.16)∂qn,m∂xi=ci,0∑l=0k′1l!(m-1-lan)!(c1,nx1+⋯+cd,nxd)l(c1,0x1+⋯+cd,0xd)m-1-lan+ci,n∑l=1k1(l-1)!(m-lan)!(c1,nx1+⋯+cd,nxd)l-1(c1,0x1+⋯+cd,0xd)m-lan=ci,0qn,m-1+ci,nqn,m-an.Case 2 (an-1≤m≤a1).
In this case,0≤s≤n-1.
By (4.11) and (4.1), it follows that m-kan-an-1<0. Thus, there exists some 0≤l0≤k-1 such that
(4.17)m-l0an-an-1≥0,m-(l0+1)an-an-1<0.
As a result, we have that 0≤m-lan<an-1 for all l0+1≤l≤k, which implies that
(4.18)qn-1,m-lan=1(m-lan)!(c1,0x1+⋯+cd,0xd)m-lan,l0+1≤l≤k.
Therefore,
(4.19)∂qn-1,m-lan∂xi={ci,0qn-1,m-1-lan,l0+1≤l≤k′;0,k′+1≤l≤k.
For all0≤l≤l0, we have an-1≤m-lan≤a1. Then our inductive hypothesis implies that
(4.20)∂qn-1,m-lan∂xi=ci,0qn-1,m-1-lan+∑j=sln-1ci,jqn-1,m-lan-aj,0≤l≤l0,
where sl is the minimal integer between 1 and n-1 such that m-lan-asl≥0. It should be noticed that s0=s.
By (4.19) and (4.20), we can deduce that
(4.21)∂qn,m∂xi=ci,0(∑l=0k′1l!(c1,nx1+⋯+cd,nxd)lqn-1,m-1-lan)+ci,n(∑l=1k1(l-1)!(c1,nx1+⋯+cd,nxd)l-1qn-1,m-lan)+∑l=0l01l!(c1,nx2+⋯+cd,nxd)l∑j=sln-1ci,jqn-1,m-lan-aj=ci,0qn,m-1+ci,nqn,m-an+∑l=0l01l!(c1,nx2+⋯+cd,nxd)l∑j=sln-1ci,jqn-1,m-lan-aj.
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 s0≤j≤n-1, let kj denote the maximum nonnegative integer satisfying kjan≤m-aj, that is,
(4.22)kjan≤m-aj,(kj+1)an>m-aj.
From (4.17), we know kn-1=l0.
Due to (4.13), we observe that
(4.23)qn,m-aj=∑l=0kj1l!(c2,nx2+⋯+cd,nxd)lqn-1,m-aj-lan,s0≤j≤n-1.
Furthermore,
(4.24)qn,m-aj=∑l=0l01l!(c1,nx2+⋯+cd,nxd)lqn-1,m-aj-lan,s0≤j≤n-1.
Here, we have used the fact that qn-1,m-aj-lan=0 for all s0≤j≤n-1, kj+1≤l≤l0.
Now, recall thatsl,0≤l≤l0 is the minimal integer between 1 and n-1 such that m-lan-asl≥0, then
(4.25)∑j=s0n-1ci,jqn-1,m-aj-lan=∑j=sln-1ci,jqn-1,m-aj-lan,0≤l≤l0.
According to (4.24), (4.25), and the fact s=s0, we find that
(4.26)∑j=sn-1ci,jqn,m-aj=∑j=s0n-1ci,j∑l=0l01l!(c1,nx2+⋯+cd,nxd)lqn-1,m-aj-lan=∑l=0l01l!(c1,nx2+⋯+cd,nxd)l(∑j=sln-1ci,jqn-1,m-aj-lan).
Consequently,
(4.27)∂qn,m∂xi=ci,0qn,m-1+∑j=snci,jqn,m-aj,an-1≤m≤a1.Proposition 4.2.
Leta=(a0,a1,…,an), ci=(ci,0,ci,1,…,ci,n) with 1≤i≤d, and let qn,m with 0≤m≤a1 be as above, h a nonzero number in 𝔽. Then for arbitrary p∈𝔽[x] and ξ∈𝔽d:
(4.28)1m!hm∑0≤r≤m(-1)m-r(mr)p(ξ+(∑j=0nc1,j(rh)aj,…,∑j=0ncd,j(rh)aj))=(qn,m(D)p)(ξ)+O(h).Proof.
Applying Taylor Formulas, we obtain(4.29)p(ξ+(∑j=0nc1,j(rh)aj,…,∑j=0ncd,j(rh)aj))=∑k=0∞∑‖γ1‖1+⋯+‖γd‖1=k(rh)∑1≤i≤d,0≤j≤najγi,jc1γ1⋯cdγdγ1!⋯γd!∂kp(ξ)∂x1‖γ1‖1⋯∂xd‖γd‖1,
where ciγi=∏j=0nci,jγi,j, γi=(γi,0,…,γi,n)∈ℕn+1,1≤i≤d.
By means of the mapτ defined as in (4.3), the above equality can be rewritten as
(4.30)p(ξ+(∑j=0nc1,j(rh)aj,…,∑j=0ncd,j(rh)aj))=∑l=0m∑τ(γ1,…,γd)=l(rh)lc1γ1⋯cdγdγ1!⋯γd!∂‖γ1‖1+⋯+‖γd‖1p(ξ)∂x1‖γ1‖1⋯∂xd‖γd‖1+O(hm+1),
where O(hm+1) is a polynomial in h.
Finally, by Lemma2.1, we can conclude that
(4.31)∑0≤r≤m(-1)m-r(mr)p(ξ+(∑j=0nc1,j(rh)aj,…,∑j=0ncd,j(rh)aj))=m!hm∑τ(γ1,…,γd)=mc1γ1…cdγdγ1!⋯γd!∂‖γ1‖1+⋯+∥γd∥1p(ξ)∂x1‖γ1‖1⋯∂xd‖γd‖1+O(hm+1).


## 5. Main Theorem

Letξ(1),…,ξ(μ),ξ(μ+1),…,ξ(μ+ν)∈𝔽d be distinct points. For each 1≤k≤μ, assume that 𝔡(k)⊂ℕd is a lower set, and ρ(k)=(ρ1(k),…,ρd(k))∈(𝔽d)d satisfies the conditions of Section 3. Likewise, for each 1≤l≤ν, assume that a(l)=(a0(l),a1(l),…,an(l)(l)), ci(l)=(ci,0(l), ci,1(l),…,ci,n(l)(l)), 1≤i≤d, and qn(l),m(l), 0≤m≤a1(l) satisfy the conditions of Section 4.Theorem 5.1.
With the notation above, letP be an ideal projector with
(5.1)ranP′=spanF{δξ(k)Dρ(k)α,δξ(μ+l)qn(l),m(l)(D):α∈d(k),1≤k≤μ,0≤m≤a1(l),1≤l≤ν},
and let Ph be a Lagrange projector with
(5.2)ranPh′=spanF{δξ(k)+h∑i=1dαiρi(k),δξ(μ+l)+ϕ(l)(mh):α∈d(k),1≤k≤μ,0≤m≤a1(l),1≤l≤ν},
where h∈𝔽∖{0} and
(5.3)ϕ(l)(mh)=(∑j=0n(l)c1,j(l)(mh)aj(l),∑j=0n(l)c2,j(l)(mh)aj(l),…,∑j=0n(l)cd,j(l)(mh)aj(l)).
Then, the following statements hold:(i)
there exists a positiveη∈𝔽 such that(5.4)ranPh=ranP,∀0<|h|<η;(ii)
P is the pointwise limit of Ph,0<|h|<η, as h tends to zero.Proof.
Firstly, one can easily verify that(5.5)λ=(δξ(k)Dρ(k)α,δξ(μ+l)qn(l),m(l)(D):α∈d(k),1≤k≤μ,0≤m≤a1(l),1≤l≤ν)∈(F[x]′)s,λh=(δξ(k)+h∑i=1dαiρi(k),δξ(μ+l)+ϕ(l)(mh):α∈d(k),1≤k≤μ,0≤m≤a1(l),1≤l≤ν)∈(F[x]′)s,
form 𝔽-bases for ranP′ and ranPh′, respectively, where
(5.6)s=∑k=1μ#d(k)+∑k=1νa1(k)+ν.
Provided that the entries of λ and λh are arranged in the same order, namely, for arbitrary fixed 1≤k≤μ, α∈𝔡(k), the corresponding entries of λ and λh are in the same position, the same as for arbitrary fixed 1≤l≤ν, 0≤m≤a1(l). We denote λ=(λ1,…,λs) and λh=(λh,1,…,λh,s), respectively.
Letq=(q1,q2,…,qs) be an 𝔽-basis for ranP, For convenience, we introduce two s×s matrices:
(5.7)λTq:=(λiqj)1≤i,j≤s,λhTq:=(λh,iqj)1≤i,j≤s
and for arbitrary f∈𝔽[x], vectors:
(5.8)λTf:=(λif)1≤i≤s,λhTf:=(λh,if)1≤i≤s.
For arbitraryp∈𝔽[x], we have the following facts according to equality (3.5) and (4.28). (1)
For fixed1≤k≤μ and α∈𝔡(k), (Dρ(k)αp)(ξ(k)) can be linearly expressed by
(5.9){p(ξ(k)+h∑i=1dβiρi(k)):β∈d(k)}∪{O(h)},
since 𝔡(k) is lower, and moreover, the linear combination coefficient of each p(ξ(k)+h∑i=1dβiρi(k)) is independent of p∈𝔽[x].(2)
For fixed1≤l≤ν and 0≤m≤a1(l), (qn(l),m(l)(D)p)(ξ(μ+l)) can be linearly expressed by
(5.10){p(ξ(μ+l)+ϕ(l)(rh)):0≤r≤m}∪{O(h)}.
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 matrixT such that
(5.11)[λhTq̂∣λhTf̂]:=T[λhTq∣λhTf]=[λTq∣λTf]+[Eh∣ϵh],
where each entry of [Eh∣ϵh] has the same order as h. As a consequence, the linear systems:
(5.12)(λhTq̂)x=λhTf̂,(λhTq)x=λhTf
are equivalent, namely, they have the same set of solutions.(i)
From (5.11), it follows that each entry of matrix λhTq̂ converges to its corresponding entry of matrix λTq as h tends to zero, which implies that
(5.13)lim⁡h→0det⁡(λhTq̂)=det⁡(λTq).
Since det(λTq)≠0, there exists η>0 such that
(5.14)det⁡(λhTq̂)≠0,0<|h|<η.
Notice that (5.11) directly leads to rank(λhTq̂)=rank(λhTq):
(5.15)ranPh=spanFq,0<|h|<η,
follows, that is, q forms an 𝔽-basis for ranPh. Since q is also an 𝔽-basis for ranP, we have
(5.16)ranP=ranPh,0<|h|<η.(ii)
Suppose thatx̃ and x0 be the unique solutions of nonsingular linear systems:
(5.17)(λhTq)x=λhTf,(5.18)(λTq)x=λTf,
respectively, where f∈𝔽[x] and 0<|h|<η. It is easy to see that(5.19)Phf=qx̃,Pf=qx0.
Notice that, as h→0, P is the pointwise limit of Ph if and only if Pf is the coefficient-wise limit of Phf 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,
(5.20)lim⁡h→0x̃=x0.
By (5.11), the linear system:
(5.21)(λhTq̂)x=λhTf̂
can be rewritten as
(5.22)(λTq+Eh)x=(λTf+ϵh).
Since system (5.21) is equivalent to system (5.17), x̃ 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)‖x̃-x0‖≤‖(λTq)-1‖‖ϵh-Ehx0‖+O(h2).
Since each component of vector ϵh-Ehx0 has the same order as h, it follows that limh→0∥x̃-x0∥=0, or, equivalently, limh→0x̃=x0, 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 ofD-invariant polynomial subspaces into a sequence of Lagrange projectors.



---

*Source: 359069-2012-06-11.xml*