The Discretization for a Special Class of Ideal Projectors

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 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,…,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:_𝝃∈Ξ𝛿_𝝃𝒬_𝝃(𝐷),(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.

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 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=(𝛼₁,…,𝛼_𝑑). For arbitrary 𝜶∈ℕ^𝑑, we define 𝜶!=𝛼₁!⋯𝛼_𝑑!.

Henceforward, ≤ will denote the usual product order on ℕ^𝑑, that is, for arbitrary 𝜶, 𝜷∈ℕ^𝑑, 𝜶≤𝜷 if and only if 𝛼_𝑖≤𝛽_𝑖,𝑖=1,…,𝑑. A finite subset 𝒜⊂ℕ^𝑑 is lower if for every 𝜶∈𝒜, 𝟎≤𝜷≤𝜶 implies 𝜷∈𝒜.

A monomial 𝐱^𝜶∈𝔽[𝐱] is a power product of the form 𝑥^𝛼1₁⋯𝑥^𝛼𝑑_𝑑 with 𝜶∈ℕ^𝑑. Thus, a polynomial 𝑝 in 𝔽[𝐱] can be expressed as𝑝=_{𝜶∈ℕ𝑑}𝑝̂(𝜶)𝐱^𝜶,(2.1) with 𝑝(̂𝜶)∈𝔽 nonzero. For a polynomial 𝑝(𝐱) as in (2.1), we write𝑓⟼𝑝(𝐷)∶_{𝜶∈ℕ𝑑}𝔽[𝐱]⟶𝔽[𝐱]𝑝̂(𝜶)𝜕^‖𝜶‖1𝑓𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑑_𝑑,(2.2) for its associated differential operator. For a finite-dimensional polynomial subspace 𝒬, we define𝒬(𝐷)∶=span_𝔽{𝑞(𝐷)∶𝑞∈𝒬}.(2.3) 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 asran𝑃∶={k𝑔er∈𝑃𝔽∶[=𝐱]{∶𝑔𝑔∈=𝔽𝑃[𝐱𝑓]∶fo𝑃r𝑔so=m0e}𝑓.∈𝔽[𝐱]},(2.4) Let 𝑃^ be the dual projector of 𝑃, then ran𝑃^ is the set of interpolation conditions matched by 𝑃. If 𝐪=(𝑞₁,…,𝑞_𝑠) is an 𝔽-basis for ran𝑃 and 𝝀=(𝜆₁,…,𝜆_𝑠) an 𝔽-basis for ran𝑃^, then their Gram matrix: 𝝀^𝑇𝐪∶=𝜆_𝑖𝑞_𝑗_{1≤𝑖,𝑗≤𝑠}(2.5) is invertible.

Lemma 2.1 (see [15, page 90]). Let 𝑛 and 𝑚 be arbitrary nonnegative integers satisfying 𝑛≤𝑚. Then ^𝑚_𝑟=0(−1)^𝑚−𝑟⎛⎜⎜⎝𝑚𝑟⎞⎟⎟⎠𝑟_𝑛=𝑚0,!,𝑛0=≤𝑚𝑛;<𝑚.(2.6)

3. The First Class of 𝐷-Invariant Polynomial Subspaces

Let 𝜶=(𝛼₁,…,𝛼_𝑑)∈ℕ^𝑑 and 𝝆=(𝝆₁,𝝆₂,…,𝝆_𝑑)∈(𝔽^𝑑)^𝑑 in which𝝆₁=𝜌_1,1,…,𝜌_1,𝑑,𝝆₂=𝜌_2,1,…,𝜌_2,𝑑,…,𝝆_𝑑=𝜌_𝑑,1,…,𝜌_𝑑,𝑑(3.1) are 𝔽-linearly independent unit vectors. Then we define a differential operator 𝐷^𝜶𝝆∶𝔽[𝐱]→𝔽[𝐱] by the formula: 𝐷^𝜶𝝆𝑓=𝜕^‖𝜶‖1𝑓𝜕𝝆^𝛼1₁⋯𝜕𝝆^𝛼𝑑_𝑑.(3.2)

Lemma 3.1. Let 𝜶=(𝛼₁,…,𝛼_𝑑)∈ℕ^𝑑, 𝝆=(𝝆₁,𝝆₂,…,𝝆_𝑑)∈(𝔽^𝑑)^𝑑 be as above, and let 𝑝=^𝑑_𝑖=1𝝆_𝑖⋅𝐱^𝛼𝑖,(3.3) where ⋅ denotes the Euclidean product. Then 𝑝(𝐷)=𝐷^𝜶𝝆.

Given a lower set 𝔡⊂ℕ^𝑑 and 𝝆=(𝝆₁,𝝆₂,…,𝝆_𝑑)∈(𝔽^𝑑)^𝑑 as above, then the subspace:𝒬=span_𝔽^𝑑_𝑖=1𝝆_𝑖⋅𝐱^𝛼𝑖∶𝜶∈𝔡(3.4) is obviously 𝐷-invariant. The following proposition discretizes this functional subspace 𝛿_𝝃𝒬(𝐷) into a sequence of evaluation functional subspaces.

Proposition 3.2. Let 𝝆=(𝝆₁,𝝆₂,…,𝝆_𝑑)∈(𝔽^𝑑)^𝑑 be as above, ℎ a nonzero number in 𝔽, and let 𝜶∈ℕ^𝑑, 𝝃∈𝔽^𝑑. Then for arbitrary 𝑝∈𝔽[𝐱]1ℎ^‖𝜶‖1_{𝟎≤𝜷≤𝜶}(−1)^{‖𝜶−𝜷‖1}⎛⎜⎜⎝𝜶𝜷⎞⎟⎟⎠𝑝𝝃+ℎ^𝑑_𝑖=1𝛽_𝑖𝝆_𝑖=𝐷^𝜶𝝆𝑝(𝝃)+𝑂(ℎ),(3.5) where ^𝜶𝜷=∏^𝑑𝑖=1^𝛼𝑖_𝛽𝑖, and the remainder 𝑂(ℎ) is a polynomial in ℎ.

Proof. For convenience, let 𝜑=_{𝟎≤𝜷≤𝜶}(−1)^{‖𝜶−𝜷‖1}⎛⎜⎜⎝𝜶𝜷⎞⎟⎟⎠𝑝𝝃+ℎ^𝑑_𝑘=1𝛽_𝑘𝝆_𝑘.(3.6) Then we can obtain that for 𝑚=0,…,‖𝜶‖₁: 𝑑^𝑚𝜑(0)𝑑ℎ^𝑚=_{‖‖𝛾1‖‖1‖‖𝜸+⋯+𝑑‖‖𝟏=𝑚}𝜌^𝛾1,1_1,1⋯𝜌^𝛾1,𝑑_𝑑,1⋯𝜌^𝛾𝑑,1_1,𝑑⋯𝜌^{𝛾𝑑,𝑑}_𝑑,𝑑𝜕^𝑚𝑝(𝝃)𝜕𝑥^‖𝜸1‖1₁…𝜕𝑥^{‖𝜸𝑑‖𝟏}_𝑑×⎛⎜⎜⎝𝛾_1,1𝑚⋯𝛾_1,𝑑,…,𝛾_𝑑,1,…,𝛾_𝑑,𝑑⎞⎟⎟⎠^𝑑_𝑗=1⎛⎜⎜⎝^𝛼𝑗_𝛽_𝑗=0(−1)^{𝛼𝑗−𝛽𝑗}⎛⎜⎜⎝𝛼_𝑗𝛽_𝑗⎞⎟⎟⎠𝛽_{∑𝑑𝑖=1𝛾𝑖,𝑗𝑗}⎞⎟⎟⎠,(3.7) with 𝜸_𝑖=(𝛾_𝑖,1,…,𝛾_𝑖,𝑑)∈ℕ^𝑑, 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 some 1≤𝑗≤𝑑 such that ∑^𝑑𝑖=1𝛾_𝑖,𝑗<𝛼_𝑗. By Lemma 2.1, it follows for such 𝑗: ^𝛼^𝑗_𝛽𝑗=0(−1)^{𝛼𝑗−𝛽𝑗}⎛⎜⎜⎝𝛼_𝑗𝛽_𝑗⎞⎟⎟⎠𝛽_{∑𝑑𝑖=1𝛾𝑖,𝑗𝑗}=0.(3.8) Thus, for all 0≤𝑚≤‖𝜶‖₁−1, 𝑑^𝑚𝜑(0)/𝑑ℎ^𝑚=0.Case 2 (𝑚=‖𝜶‖₁). In this case, if there exists some 1≤𝑗≤𝑑 such that ∑^𝑑𝑖=1𝛾_𝑖,𝑗>𝛼_𝑗, then there must exist another 1≤𝑗′≤𝑑, such that ∑^𝑑𝑖=1𝛾_𝑖,𝑗′<𝛼_𝑗′. Hence, when 𝑚=‖𝜶‖₁, ^𝑑_𝑗=1⎛⎜⎜⎝^𝛼𝑗_𝛽_𝑗=0(−1)^{𝛼𝑗−𝛽𝑗}⎛⎜⎜⎝𝛼_𝑗𝛽_𝑗⎞⎟⎟⎠𝛽_{∑𝑑𝑖=1𝛾𝑖,𝑗𝑗}⎞⎟⎟⎠≠0(3.9) if and only if for all 1≤𝑗≤𝑑, ∑^𝑑𝑖=1𝛾_𝑖,𝑗=𝛼_𝑗. Combining this with the fact: ^𝛼^𝑗_𝛽𝑗=0(−1)^{𝛼𝑗−𝛽𝑗}⎛⎜⎜⎝𝛼_𝑗𝛽_𝑗⎞⎟⎟⎠𝛽_𝛼𝑗𝑗=𝛼_𝑗!,1≤𝑗≤𝑑,(3.10) we can conclude that 𝑑^‖𝜶‖1𝜑(0)𝑑ℎ^‖𝜶‖1=‖𝜶‖₁!_{𝛾1,1+⋯+𝛾𝑑,1=𝛼1}⋯_{𝛾1,𝑑+⋯+𝛾𝑑,𝑑=𝛼𝑑}𝜌^𝛾1,1_1,1⋯𝜌^𝛾1,𝑑_𝑑,1⋯𝜌^𝛾𝑑,1_1,𝑑⋯𝜌^{𝛾𝑑,𝑑}_𝑑,𝑑×⎛⎜⎜⎝𝛼¹𝛾_1,1,…,𝛾_𝑑,1⎞⎟⎟⎠⋯⎛⎜⎜⎝𝛼_𝑑𝛾_1,𝑑,…,𝛾_𝑑,𝑑⎞⎟⎟⎠𝜕^‖𝜶‖₁𝑝(𝝃)𝜕𝑥^‖𝜸1‖1₁⋯𝜕𝑥^{‖𝜸𝑑‖𝟏}_𝑑.(3.11) That is 𝑑^‖𝜶‖1𝜑(0)𝑑ℎ^‖𝜶‖1=‖𝜶‖₁!𝐷^𝜶𝝆𝑝(𝝃).(3.12) In sum, we have deduced that 𝑑^𝑚𝜑(0)𝑑ℎ^𝑚=0,0≤𝑚≤‖𝜶‖¹‖𝜶‖₁−1;!𝐷^𝜶𝝆𝑝(𝝃),𝑚=‖𝜶‖₁,(3.13) which implies that 𝜑=ℎ^‖𝜶‖1𝐷^𝜶𝝆𝑝(𝝃)+𝑂ℎ^{‖𝜶‖1+1}.(3.14)

4. The Second Class of 𝐷-Invariant Polynomial Subspaces

Let 𝐚=(𝑎₀,𝑎₁,…,𝑎_𝑛) with 𝑛≥1 be an 𝑛+1-tuple of positive integers satisfying𝑎₀=1,𝑎₁>⋯>𝑎_𝑛≥2,(4.1) and let 𝐜₁=𝑐_1,0,𝑐_1,1,…,𝑐_1,𝑛,…,𝐜_𝑑=𝑐_𝑑,0,𝑐_𝑑,1,…,𝑐_𝑑,𝑛∈𝔽^𝑛+1(4.2) satisfying that 𝑐_1,0,𝑐_2,0,…,𝑐_𝑑,0 are not all zero. Then it is clear that𝜏𝜸₁,…,𝜸_𝑑=^𝑛_𝑗=0𝑎_𝑗𝑑_𝑖=1𝛾_𝑖,𝑗,(4.3) with 𝜸_𝑖=(𝛾_𝑖,0,𝛾_𝑖,1,…,𝛾_𝑖,𝑛)∈ℕ^𝑛+1 defines a map 𝜏∶(ℕ^𝑛+1)^𝑑→ℕ.

Proposition 4.1. Let 𝐚=(𝑎₀,𝑎₁,…,𝑎_𝑛), 𝐜_𝑖=(𝑐_𝑖,0,𝑐_𝑖,1,…,𝑐_𝑖,𝑛), 1≤𝑖≤𝑑, and the map 𝜏 be as above. Let 𝑞_𝑛,𝑚, 𝑚=0,1,…,𝑎₁ be polynomials defined by 𝑞_𝑛,𝑚=_{𝜏(𝜸1,…,𝜸𝑑)=𝑚}𝐜^𝜸1₁⋯𝐜^𝜸𝑑_𝑑𝜸₁!⋯𝜸_𝑑!𝑥^{‖𝜸1‖11}⋯𝑥^{‖𝜸𝑑‖1}_𝑑,(4.4) with 𝜸_𝑖=(𝛾_𝑖,0,𝛾_𝑖,1,…,𝛾_𝑖,𝑛)∈ℕ^𝑛+1, 𝐜^𝜸𝑖_𝑖=∏^𝑛𝑗=0𝑐^{𝛾𝑖,𝑗}_𝑖,𝑗, and let 𝒬 be a polynomial subspace defined by 𝒬=span_𝔽𝑞_𝑛,𝑚∶0≤𝑚≤𝑎₁.(4.5) Then, the following hold:(i)dim𝒬=𝑎₁+1;(ii)𝒬 is a 𝐷-invariant polynomial subspace.

Proof. (i) We assume that there exist 𝑘₀,𝑘₁,…,𝑘_𝑎1−1,𝑘_𝑎1∈𝔽 such that 𝑘₀𝑞_𝑛,0+𝑘₁𝑞_𝑛,1+⋯+𝑘_𝑎1−1𝑞_{𝑛,𝑎1−1}+𝑘_𝑎1𝑞_𝑛,𝑎1=0.(4.6) Since there exists some 1≤𝑖₀≤𝑑 such that 𝑐_𝑖0,0≠0, then 𝑥^𝑚𝑖₀ 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^𝑎1_𝑖0 belongs to the support of 𝑞_𝑛,𝑎1, while for all 0≤𝑚≤𝑎₁−1, 𝑥^𝑎1_𝑖0 cannot belong to the support of 𝑞_𝑛,𝑚. Therefore 𝑘_𝑎1=0, which implies that 𝑘₀𝑞_𝑛,0+𝑘₁𝑞_𝑛,1+⋯+𝑘_𝑎1−1𝑞_{𝑛,𝑎1−1}=0.(4.7)
Arguing for 𝑚=𝑎₁−1,𝑎₁−2,…,1,0 as for 𝑚=𝑎₁, we get 𝑘_𝑎1−1=0,𝑘_𝑎1−2=0,…,𝑘₁=0,𝑘₀=0, successively. That is to say, the family of polynomials 𝑞_𝑛,𝑚,0≤𝑚≤𝑎₁ is 𝔽-linearly independent. So dim𝒬=𝑎₁+1.
(ii) If 1≤𝑚<𝑎_𝑛, then 𝜏(𝜸₁,…,𝜸_𝑑)=𝑚 implies that 𝛾_𝑖,𝑗=0 with 1≤𝑖≤𝑑,1≤𝑗≤𝑛. Thus, for all 1≤𝑚<𝑎_𝑛: 𝑞_𝑛,𝑚=_{𝛾1,0+⋯+𝛾𝑑,0=𝑚}𝑐^𝛾1,0_1,0⋯𝑐^𝛾𝑑,0_𝑑,0𝛾_1,0!⋯𝛾_𝑑,0!𝑥^𝛾1,01⋯𝑥^𝛾𝑑,0_𝑑=1𝑚!𝑐^1,0𝑥₁+𝑐_2,0𝑥₂+⋯+𝑐_𝑑,0𝑥_𝑑^𝑚.(4.8) Consequently, 𝜕𝑞_𝑛,𝑚𝜕𝑥_𝑖=𝑐_𝑖,0𝑞_{𝑛,𝑚−1},1≤𝑚<𝑎_𝑛.(4.9)
Next, let 𝑚 be an arbitrary integer satisfying 𝑎_𝑛≤𝑚≤𝑎₁, and 𝑠 the minimum integer between 1 and 𝑛 such that 𝑚−𝑎_𝑠≥0. We claim 𝜕𝑞_𝑛,𝑚𝜕𝑥_𝑖=𝑐_𝑖,0𝑞_{𝑛,𝑚−1}+^𝑛_𝑗=𝑠𝑐_𝑖,𝑗𝑞_{𝑛,𝑚−𝑎𝑗},𝑎_𝑛≤𝑚≤𝑎₁.(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, 𝑘𝑎_𝑛≤𝑚,(𝑘+1)𝑎_𝑛>𝑚(4.11) and 𝑘^ the maximum nonnegative integer such that 𝑘′𝑎_𝑛≤𝑚−1. From the definition of 𝑞_𝑛,𝑚, we obtain that 𝑞_𝑛,𝑚=^𝑘_𝑙=0⎛⎜⎜⎝_{𝜏𝛾1,0,…,𝛾1,𝑛−1𝛾,…,𝑑,0,…,𝛾𝑑,𝑛−1=𝑚−𝑙𝑎𝑛}𝑐^𝛾1,0_1,0⋯𝑐^{𝛾1,𝑛−1}_1,𝑛−1⋯𝑐^𝛾𝑑,0_𝑑,0⋯𝑐^{𝛾𝑑,𝑛−1}_{𝑑,𝑛−1}×𝑥_{𝛾1,0+⋯+𝛾1,𝑛−11}𝛾_1,0!⋯𝛾_1,𝑛−1!⋯𝛾_𝑑,0!⋯𝛾_{𝑑,𝑛−1}!⋯𝑥^{𝛾𝑑,0+⋯+𝛾𝑑,𝑛−1}_𝑑⎞⎟⎟⎠⎛⎜⎜⎝_{𝛾1,𝑛+⋯+𝛾𝑑,𝑛=𝑙}𝑐^𝛾1,𝑛_1,𝑛⋯𝑐^{𝛾𝑑,𝑛}_𝑑,𝑛𝑐_1,𝑛!⋯𝑐_𝑑,𝑛!𝑥^𝛾1,𝑛1⋯𝑥^{𝛾𝑑,𝑛}_𝑑⎞⎟⎟⎠.(4.12) That is 𝑞_𝑛,𝑚=^𝑘_𝑙=01𝑙!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛},(4.13) 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, 𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛}=1𝑚−𝑙𝑎_𝑛!𝑐^1,0𝑥₁+⋯+𝑐_𝑑,0𝑥_𝑑^{𝑚−𝑙𝑎𝑛},0≤𝑙≤𝑘.(4.14) By (4.13), we get 𝑞_𝑛,𝑚=^𝑘_𝑙=01𝑙!𝑚−𝑙𝑎_𝑛!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑐_1,0𝑥₁+⋯+𝑐_𝑑,0𝑥_𝑑^{𝑚−𝑙𝑎𝑛}.(4.15) So we obtain that 𝜕𝑞_𝑛,𝑚𝜕𝑥_𝑖=𝑐_𝑘𝑖,0^′_𝑙=01𝑙!𝑚−1−𝑙𝑎_𝑛!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑐_1,0𝑥₁+⋯+𝑐_𝑑,0𝑥_𝑑^{𝑚−1−𝑙𝑎𝑛}+𝑐_{𝑘𝑖,𝑛}_𝑙=11(𝑙−1)!𝑚−𝑙𝑎_𝑛!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙−1𝑐_1,0𝑥₁+⋯+𝑐_𝑑,0𝑥_𝑑^{𝑚−𝑙𝑎𝑛}=𝑐_𝑖,0𝑞_{𝑛,𝑚−1}+𝑐_𝑖,𝑛𝑞_{𝑛,𝑚−𝑎𝑛}.(4.16)Case 2 (𝑎_𝑛−1≤𝑚≤𝑎₁). 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≤𝑙₀≤𝑘−1 such that 𝑚−𝑙₀𝑎_𝑛−𝑎_𝑛−1≥0,𝑚−𝑙₀+1𝑎_𝑛−𝑎_𝑛−1<0.(4.17) As a result, we have that 0≤𝑚−𝑙𝑎_𝑛<𝑎_𝑛−1 for all 𝑙₀+1≤𝑙≤𝑘, which implies that 𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛}=1𝑚−𝑙𝑎_𝑛!𝑐^1,0𝑥₁+⋯+𝑐_𝑑,0𝑥_𝑑^{𝑚−𝑙𝑎𝑛},𝑙₀+1≤𝑙≤𝑘.(4.18) Therefore, 𝜕𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛}𝜕𝑥_𝑖=𝑐^𝑖,00,𝑞_{𝑛−1,𝑚−1−𝑙𝑎𝑛},𝑙𝑘_0+1≤𝑙≤𝑘^;+1≤𝑙≤𝑘.(4.19)
For all 0≤𝑙≤𝑙₀, we have 𝑎_𝑛−1≤𝑚−𝑙𝑎_𝑛≤𝑎₁. Then our inductive hypothesis implies that 𝜕𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛}𝜕𝑥_𝑖=𝑐_𝑖,0𝑞_{𝑛−1,𝑚−1−𝑙𝑎𝑛}+^𝑛−1_{𝑗=𝑠𝑙}𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛−𝑎𝑗},0≤𝑙≤𝑙₀,(4.20) where 𝑠_𝑙 is the minimal integer between 1 and 𝑛−1 such that 𝑚−𝑙𝑎_𝑛−𝑎_𝑠𝑙≥0. It should be noticed that 𝑠₀=𝑠.
By (4.19) and (4.20), we can deduce that 𝜕𝑞_𝑛,𝑚𝜕𝑥_𝑖=𝑐_𝑖,0⎛⎜⎜⎝^𝑘′_𝑙=01𝑙!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑞_{𝑛−1,𝑚−1−𝑙𝑎𝑛}+𝑐_𝑖,𝑛⎞⎟⎟⎠^𝑘_𝑙=11(𝑙−1)!𝑐^1,𝑛𝑥₁+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙−1𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛}+_𝑙0_𝑙=01𝑙!𝑐^1,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑛−1_{𝑗=𝑠𝑙}𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛−𝑎𝑗}=𝑐_𝑖,0𝑞_{𝑛,𝑚−1}+𝑐_𝑖,𝑛𝑞_{𝑛,𝑚−𝑎𝑛}+^𝑙0_𝑙=01𝑙!𝑐^1,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑛−1_{𝑗=𝑠𝑙}𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑙𝑎𝑛−𝑎𝑗}.(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 𝑠₀≤𝑗≤𝑛−1, let 𝑘_𝑗 denote the maximum nonnegative integer satisfying 𝑘_𝑗𝑎_𝑛≤𝑚−𝑎_𝑗, that is, 𝑘_𝑗𝑎_𝑛≤𝑚−𝑎_𝑗,𝑘_𝑗+1𝑎_𝑛>𝑚−𝑎_𝑗.(4.22) From (4.17), we know 𝑘_𝑛−1=𝑙₀.
Due to (4.13), we observe that 𝑞_{𝑛,𝑚−𝑎𝑗}=^𝑘𝑗_𝑙=01𝑙!𝑐^2,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛},𝑠₀≤𝑗≤𝑛−1.(4.23) Furthermore, 𝑞_{𝑛,𝑚−𝑎𝑗}=^𝑙0_𝑙=01𝑙!𝑐^1,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛},𝑠₀≤𝑗≤𝑛−1.(4.24) Here, we have used the fact that 𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛}=0 for all 𝑠₀≤𝑗≤𝑛−1, 𝑘_𝑗+1≤𝑙≤𝑙₀.
Now, recall that 𝑠_𝑙,0≤𝑙≤𝑙₀ is the minimal integer between 1 and 𝑛−1 such that 𝑚−𝑙𝑎_𝑛−𝑎_𝑠𝑙≥0, then ^𝑛⁻¹_𝑗=𝑠0𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛}=^𝑛−1_{𝑗=𝑠𝑙}𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛},0≤𝑙≤𝑙₀.(4.25)
According to (4.24), (4.25), and the fact 𝑠=𝑠₀, we find that ^𝑛−1_𝑗=𝑠𝑐_𝑖,𝑗𝑞_{𝑛,𝑚−𝑎𝑗}=^𝑛−1_𝑗_=𝑠0𝑐_{𝑙𝑖,𝑗}⁰_𝑙=01𝑙!𝑐^1,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛}=^𝑙0_𝑙=01𝑙!𝑐^1,𝑛𝑥₂+⋯+𝑐_𝑑,𝑛𝑥_𝑑^𝑙^𝑛−1_{𝑗=𝑠𝑙}𝑐_𝑖,𝑗𝑞_{𝑛−1,𝑚−𝑎𝑗−𝑙𝑎𝑛}.(4.26) Consequently, 𝜕𝑞_𝑛,𝑚𝜕𝑥_𝑖=𝑐_𝑖,0𝑞_{𝑛,𝑚−1}+^𝑛_𝑗=𝑠𝑐_𝑖,𝑗𝑞_{𝑛,𝑚−𝑎𝑗},𝑎_𝑛−1≤𝑚≤𝑎₁.(4.27)

Proposition 4.2. Let 𝐚=(𝑎₀,𝑎₁,…,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≤𝑎₁ be as above, ℎ a nonzero number in 𝔽. Then for arbitrary 𝑝∈𝔽[𝐱] and 𝝃∈𝔽^𝑑: 1𝑚!ℎ^𝑚_{0≤𝑟≤𝑚}(−1)^𝑚−𝑟⎛⎜⎜⎝𝑚𝑟⎞⎟⎟⎠𝑝𝝃+^𝑛_𝑗=0𝑐_1,𝑗(𝑟ℎ)^𝑎𝑗,…,^𝑛_𝑗=0𝑐_𝑑,𝑗(𝑟ℎ)^𝑎𝑗=𝑞^𝑛,𝑚(𝐷)𝑝(𝝃)+𝑂(ℎ).(4.28)

Proof. Applying Taylor Formulas, we obtain 𝑝𝝃+^𝑛_𝑗=0𝑐_1,𝑗(𝑟ℎ)^𝑎𝑗,…,^𝑛_𝑗=0𝑐_𝑑,𝑗(𝑟ℎ)^𝑎𝑗=_∞_𝑘=0_{‖‖𝜸1‖‖1‖‖𝜸+⋯+𝑑‖‖1=𝑘}(𝑟ℎ)^{∑1≤𝑖≤𝑑,0≤𝑗≤𝑛𝑎𝑗𝛾𝑖,𝑗}𝐜^𝜸1₁⋯𝐜^𝜸𝑑_𝑑𝜸₁!⋯𝜸_𝑑!𝜕^𝑘𝑝(𝝃)𝜕𝑥^‖𝜸1‖1₁⋯𝜕𝑥^{‖𝜸𝑑‖1}_𝑑,(4.29) where 𝐜^𝜸𝑖_𝑖=∏^𝑛𝑗=0𝑐^{𝛾𝑖,𝑗}_𝑖,𝑗, 𝜸_𝑖=(𝛾_𝑖,0,…,𝛾_𝑖,𝑛)∈ℕ^𝑛+1,1≤𝑖≤𝑑.
By means of the map 𝜏 defined as in (4.3), the above equality can be rewritten as 𝑝𝝃+^𝑛_𝑗=0𝑐_1,𝑗(𝑟ℎ)^𝑎𝑗,…,^𝑛_𝑗=0𝑐_𝑑,𝑗(𝑟ℎ)^𝑎𝑗=_𝑚_𝑙=0_{𝜏𝜸1,…,𝜸𝑑=𝑙}(𝑟ℎ)^𝑙𝐜^𝜸1₁⋯𝐜^𝜸𝑑_𝑑𝜸₁!⋯𝜸_𝑑!𝜕^{‖𝜸1‖1+⋯+‖𝜸𝑑‖1}𝑝(𝝃)𝜕𝑥^‖𝜸1‖1₁⋯𝜕𝑥^{‖𝜸𝑑‖1}_𝑑+𝑂ℎ^𝑚+1,(4.30) where 𝑂(ℎ^𝑚+1) is a polynomial in ℎ.
Finally, by Lemma 2.1, we can conclude that _0≤_𝑟≤𝑚(−1)^𝑚−𝑟⎛⎜⎜⎝𝑚𝑟⎞⎟⎟⎠𝑝𝝃+^𝑛_𝑗=0𝑐_1,𝑗(𝑟ℎ)^𝑎𝑗,…,^𝑛_𝑗=0𝑐_𝑑,𝑗(𝑟ℎ)^𝑎𝑗=𝑚!ℎ_𝑚_{𝜏𝜸1,…}_{,𝜸𝑑=𝑚}𝐜^𝜸1₁…𝐜^𝜸𝑑_𝑑𝜸₁!⋯𝜸_𝑑!𝜕^{‖𝜸1‖1+⋯+‖𝜸𝑑‖1}𝑝(𝝃)𝜕𝑥^‖𝜸1‖1₁⋯𝜕𝑥^{‖𝜸𝑑‖1}_𝑑+𝑂ℎ^𝑚+1.(4.31)

5. Main Theorem

Let 𝝃⁽¹⁾,…,𝝃^(𝜇),𝝃^(𝜇+1),…,𝝃^(𝜇+𝜈)∈𝔽^𝑑 be distinct points. For each 1≤𝑘≤𝜇, assume that 𝔡^(𝑘)⊂ℕ^𝑑 is a lower set, and 𝝆^(𝑘)=(𝝆^1(𝑘),…,𝝆^𝑑(𝑘))∈(𝔽^𝑑)^𝑑 satisfies the conditions of Section 3. Likewise, for each 1≤𝑙≤𝜈, assume that 𝐚^(𝑙)=(𝑎^0(𝑙),𝑎^1(𝑙),…,𝑎^{(𝑙)𝑛(𝑙)}), 𝐜^𝑖(𝑙)=(𝑐^(𝑙)𝑖,0, 𝑐^(𝑙)𝑖,1,…,𝑐^{(𝑙)𝑖,𝑛(𝑙)}), 1≤𝑖≤𝑑, and 𝑞^{(𝑙)𝑛(𝑙),𝑚}, 0≤𝑚≤𝑎^1(𝑙) satisfy the conditions of Section 4.

Theorem 5.1. With the notation above, let 𝑃 be an ideal projector with ran𝑃^=span_𝔽𝛿^𝝃(𝑘)𝐷^𝜶𝝆_(𝑘),𝛿_𝝃^(𝜇+𝑙)𝑞^{(𝑙)𝑛(𝑙),𝑚}(𝐷)∶𝜶∈𝔡^(𝑘),1≤𝑘≤𝜇,0≤𝑚≤𝑎^1(𝑙),1≤𝑙≤𝜈,(5.1) and let 𝑃_ℎ be a Lagrange projector with ran𝑃^ℎ=span_𝔽𝛿^{𝝃(𝑘)∑+ℎ𝑑𝑖=1𝛼}_𝑖^{𝝆𝑖(𝑘)},𝛿_𝝃^(𝜇+𝑙)_+𝜙^(𝑙)_(𝑚ℎ)∶𝜶∈𝔡^(𝑘),1≤𝑘≤𝜇,0≤𝑚≤𝑎^1(𝑙),1≤𝑙≤𝜈,(5.2) where ℎ∈𝔽⧵{0} and 𝜙^(𝑙)(𝑚ℎ)=^𝑛(𝑙)_𝑗=0𝑐^(𝑙)1,𝑗(𝑚ℎ)^{𝑎𝑗(𝑙)},^𝑛(𝑙)_𝑗=0𝑐^(𝑙)2,𝑗(𝑚ℎ)^{𝑎𝑗(𝑙)},…,^𝑛(𝑙)_𝑗=0𝑐^{(𝑙)𝑑,𝑗}(𝑚ℎ)^{𝑎𝑗(𝑙)}.(5.3) Then, the following statements hold:(i)there exists a positive 𝜂∈𝔽 such thatran𝑃_ℎ=ran𝑃,∀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 𝝀_ℎ𝝀=𝛿_𝝃^(𝑘)𝐷^𝜶𝝆_(𝑘),𝛿_𝝃^(𝜇+𝑙)𝑞^{(𝑙)𝑛(𝑙),𝑚}(𝐷)∶𝜶∈𝔡^(𝑘),1≤𝑘≤𝜇,0≤𝑚≤𝑎^1(𝑙),1≤𝑙≤𝜈∈𝔽[𝐱]^^𝑠,=𝛿_{𝝃(𝑘)∑+ℎ}^𝑑𝑖=1_𝛼𝑖𝝆^𝑖(𝑘),𝛿_𝝃^(𝜇+𝑙)_+𝜙^(𝑙)_(𝑚ℎ)∶𝜶∈𝔡^(𝑘),1≤𝑘≤𝜇,0≤𝑚≤𝑎^1(𝑙),1≤𝑙≤𝜈∈𝔽[𝐱]^^𝑠,(5.5) form 𝔽-bases for ran𝑃^ and ran𝑃^ℎ, respectively, where 𝑠=^𝜇_𝑘=1#𝔡^(𝑘)+^𝜈_𝑘=1𝑎^1(𝑘)+𝜈.(5.6) 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^1(𝑙). We denote 𝝀=(𝜆₁,…,𝜆_𝑠) and 𝝀_ℎ=(𝜆_ℎ,1,…,𝜆_ℎ,𝑠), respectively.
Let 𝐪=(𝑞₁,𝑞₂,…,𝑞_𝑠) be an 𝔽-basis for ran𝑃, For convenience, we introduce two 𝑠×𝑠 matrices: 𝝀^𝑇𝐪∶=𝜆_𝑖𝑞_𝑗_{1≤𝑖,𝑗≤𝑠},𝝀^𝑇ℎ𝐪∶=𝜆_ℎ,𝑖𝑞_𝑗_{1≤𝑖,𝑗≤𝑠}(5.7) and for arbitrary 𝑓∈𝔽[𝐱], vectors: 𝝀^𝑇𝑓∶=𝜆_𝑖𝑓_{1≤𝑖≤𝑠},𝝀^𝑇ℎ𝑓∶=𝜆_ℎ,𝑖𝑓_{1≤𝑖≤𝑠}.(5.8)
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^𝜶𝝆_(𝑘)𝑝)(𝝃^(𝑘)) can be linearly expressed by 𝑝𝝃^(𝑘)+ℎ^𝑑_𝑖=1𝛽_𝑖𝝆^𝑖(𝑘)∶𝜷∈𝔡^(𝑘)∪{𝑂(ℎ)},(5.9) since 𝔡^(𝑘) is lower, and moreover, the linear combination coefficient of each 𝑝(𝝃^(𝑘)+ℎ∑^𝑑𝑖=1𝛽_𝑖𝝆^𝑖(𝑘)) is independent of 𝑝∈𝔽[𝐱].(2) For fixed 1≤𝑙≤𝜈 and 0≤𝑚≤𝑎^1(𝑙), (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 𝝀^𝑇ℎ𝐪∣𝝀^𝑇ℎ𝑓∶=𝑇𝝀^𝑇ℎ𝐪∣𝝀^𝑇ℎ𝑓=𝝀^𝑇𝐪∣𝝀^𝑇𝑓+𝐸_ℎ∣𝝐_ℎ,(5.11) where each entry of [𝐸_ℎ∣𝝐_ℎ] has the same order as ℎ. As a consequence, the linear systems: 𝝀^𝑇ℎ𝐪𝐱=𝝀^𝑇ℎ𝑓,𝝀^𝑇ℎ𝐪𝐱=𝝀^𝑇ℎ𝑓(5.12) are equivalent, namely, they have the same set of solutions.(i)From (5.11), it follows that each entry of matrix 𝝀_𝑇ℎ𝐪 converges to its corresponding entry of matrix 𝝀^𝑇𝐪 as ℎ tends to zero, which implies that lim_ℎ→0det𝝀^𝑇ℎ𝐪=det𝝀^𝑇𝐪.(5.13) Since det(𝝀^𝑇𝐪)≠0, there exists 𝜂>0 such that det𝝀^𝑇ℎ𝐪≠0,0<||ℎ||<𝜂.(5.14) Notice that (5.11) directly leads to rank(𝝀^𝑇ℎ𝐪)=rank(𝝀^𝑇ℎ𝐪): ran𝑃_ℎ=span_𝔽𝐪,0<||ℎ||<𝜂,(5.15) follows, that is, 𝐪 forms an 𝔽-basis for ran𝑃_ℎ. Since 𝐪 is also an 𝔽-basis for ran𝑃, we have ran𝑃=ran𝑃_ℎ,0<||ℎ||<𝜂.(5.16)(ii) Suppose that 𝐱̃ and 𝐱_𝟎 be the unique solutions of nonsingular linear systems: 𝝀𝝀^𝑇ℎ𝐪𝐱=𝝀^𝑇ℎ𝑓,(5.17)_𝑇𝐪𝐱=𝝀^𝑇𝑓,(5.18) respectively, where 𝑓∈𝔽[𝐱] and 0<|ℎ|<𝜂. It is easy to see that𝑃_ℎ𝑓=𝐪𝐱̃,𝑃𝑓=𝐪𝐱_𝟎.(5.19) 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_ℎ𝑓 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→0𝐱̃=𝐱_𝟎.(5.20) By (5.11), the linear system: 𝝀^𝑇ℎ𝐪𝐱=𝝀^𝑇ℎ𝑓(5.21) can be rewritten as 𝝀^𝑇𝐪+𝐸_ℎ𝐱=𝝀^𝑇𝑓+𝝐_ℎ.(5.22) 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 ‖‖𝐱̃−𝐱_𝟎‖‖≤‖‖𝝀^𝑇𝐪⁻¹‖‖‖‖𝝐_ℎ−𝐸_ℎ𝐱_𝟎‖‖+𝑂ℎ².(5.23) Since each component of vector 𝝐_ℎ−𝐸_ℎ𝐱_𝟎 has the same order as ℎ, it follows that lim_ℎ→0‖𝐱̃−𝐱_𝟎‖=0, or, equivalently, lim_ℎ→0𝐱̃=𝐱_𝟎, 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 𝐷-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.