On existence of certain error formulas for a special class of ideal projectors

Zhe Li Shugong Zhang sgzh@jlu.edu.cn Tian Dong School of Mathemathics, Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, PR China

Abstract

In this paper, we focus on a special class of ideal projectors. With the aid of algebraic geometry, we prove that for this special class of ideal projectors, there exist “good” error formulas as defined by C. de Boor. Furthermore, we completely analyze the properties of the interpolation conditions matched by this special class of ideal projectors, and show that the ranges of this special class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.

keywords:

Ideal projectors , Error formulas , Reduced Gröbner bases , Minimal degree interpolation spaces

MSC:

65D05 , 41A80 , 13P10

1 Introduction

The problem of polynomial interpolation is to construct a function $p$ belonging to a finite-dimensional subspace of $\mathbb{F}[\bm{x}]$ that agrees with another given function $f$ on a set of interpolation conditions, where $\mathbb{F}[\bm{x}]:=\mathbb{F}[x_{1},\ldots,x_{d}]$ denotes the polynomial ring in $d$ variables over the field $\mathbb{F}$ . If there exists a unique solution of the interpolation problem for every $f$ , we say that the interpolation problem is poised. It’s important to make the comment that $\mathbb{F}$ is a field of characteristic zero in this paper, for example $\mathbb{F}=\mathbb{Q},\mathbb{R},\mathbb{C}$ .

Error formulas for polynomial interpolation give explicit representations for the interpolation error. C. de Boor [3] derived a formula for the interpolation error. In terms of Newton fundamental polynomials, T. Sauer and Y. Xu [11, 12] presented Sauer-Xu error formulas for polynomial interpolation whose interpolation conditions have certain constraints. Afterward, C. de Boor [4] discussed the error formulas in tensor-product and Chung-Yao interpolation. In 1998, S. Waldron [18] investigated the error in linear interpolation at the vertices of a simplex.

As an elegant form of multivariate approximate, ideal interpolation provides a natural link between polynomial interpolation and algebraic geometry. According to G. Birkhoff’s definition [2], a linear idempotent operator $P$ on $\mathbb{F}[\bm{x}]$ is called an ideal projector if $\mathrm{ker}P$ is an ideal. In the theory of ideal interpolation, we are interested in finite-rank ideal projectors. A finite-rank ideal projector refers to the ideal projector whose range is a finite-dimensional subspace of $\mathbb{F}[\bm{x}]$ . As mentioned by Carl de Boor, one reason for choosing ideal interpolation in the first place is the resulting possibility of writing the error formulas as in the following definition.

Definition 1.

([5]) Let $P$ be an ideal projector and $\{h_{1},\ldots,h_{m}\}$ be an ideal basis for $\mathrm{ker}P$ . We say that the basis $\{h_{1},\ldots,h_{m}\}$ admits a “good” error formula if there exist homogeneous polynomials $H_{j}$ and linear operators $C_{j}:\mathbb{F}[\bm{x}]\rightarrow\mathbb{F}[\bm{x}],j=1,\ldots,m$ such that for all $f\in\mathbb{F}[\bm{x}]$ ,

H_{j}(D)h_{k}=\delta_{j,k}\mbox{~~~~~for~all~}j,k=1,\ldots,m

and

f-Pf=\sum_{j=1}^{m}C_{j}(H_{j}(D)f)h_{j},

where $H_{j}(D)$ will be defined in Section 2.

We say that $P$ has a “good” error formula if there exists an ideal basis $\{h_{1},\ldots,h_{m}\}$ for $\mathrm{ker}P$ that admits a “good” error formula.

It is no surprise that every ideal projector in univariate polynomial ring has a “good” error formula [13, 14]. When we turn to multivariate interpolation, things change greatly. C. de Boor [4] proved the existence of “good” error formulas for tensor-product and Chung-Yao interpolation. However, B. Shekhtman [15] showed that for a specific form of ideal interpolation by linear polynomials in two variables, such a “good” error formula doesn’t exist. Hence, the study of the type of ideal projectors with “good” error formulas is a rather complicated topic.

In this paper, we deal with a special class of ideal projectors, and prove the existence of “good” error formulas for this class of ideal projectors. It should be noted that the construction of the linear operators $C_{j},1\leq j\leq m$ , as in Definition 1 will be the subject of our future work. Moreover, we discuss the properties of the interpolation conditions matched by this special class of ideal projectors. The main results of this paper will be put in Section 3. The next section, Section 2, is devoted as a preparation for this paper.

2 Preliminaries

In this section, we will introduce some notation and recall some basic facts about ideal interpolation and algebraic geometry. For more details, we refer the reader to [5, 14, 7, 1].

Throughout the paper, we use $\mathbb{N}$ to stand for the set of nonnegative integers, and use boldface letters to express tuples and denote their entries by the same letter with subscripts, for example, $\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{d})$ . For arbitrary $\bm{\alpha}\in\mathbb{N}^{d}$ , we define $\bm{\alpha}!=\alpha_{1}!\ldots\alpha_{d}!$ .

A monomial $\bm{x}^{\bm{\alpha}}$ is a power product of the form $x_{1}^{\alpha_{1}}\ldots x_{d}^{\alpha_{d}}$ with ${\bm{\alpha}}\in\mathbb{N}^{d}$ . We denote by $\mathbb{T}(\bm{x}):=\mathbb{T}(x_{1},\ldots,x_{d})$ the set of all monomials in $\mathbb{F}[\bm{x}]$ . For a polynomial $f(\bm{x})=\sum_{\bm{\alpha}\in\mathbb{N}^{d}}c_{\bm{\alpha}}{\bm{x}}^{{\bm{\alpha}}}\in\mathbb{F}[\bm{x}]$ with $0\neq c_{\bm{\alpha}}\in\mathbb{F}$ , we write the associated differential operator for $f$ in the form

f(D)=\sum\limits_{\bm{\alpha}\in\mathbb{N}^{d}}c_{\bm{\alpha}}D^{\bm{\alpha}},

where

D^{\bm{\alpha}}=\frac{\partial^{\alpha_{1}+\cdots+\alpha_{d}}}{\partial x_{1}^{\alpha_{1}}\ldots\partial x_{d}^{\alpha_{d}}}.

Henceforward, we use $\leq$ to denote the usual product order on $\mathbb{N}^{d}$ . For $\bm{\alpha},\bm{\beta}\in\mathbb{N}^{d}$ , $\bm{\alpha}\leq\bm{\beta}$ if and only if $\alpha_{i}\leq\beta_{i},i=1,\ldots,d$ . In particular, $\bm{\alpha}<\bm{\beta}$ if and only if $\bm{\alpha}\leq\bm{\beta}$ and $\bm{\alpha}\neq\bm{\beta}$ . A finite subset $\mathcal{A}\subset\mathbb{N}^{d}$ is lower if for every $\bm{\alpha}\in\mathcal{A}$ , $\bm{0}\leq\bm{\beta}\leq\bm{\alpha}$ implies $\bm{\beta}\in\mathcal{A}$ .

A finite monomial set $\mathcal{O}\subset\mathbb{T}(\bm{x})$ is called an order ideal if it is closed under monomial division, namely $\bm{t}\in\mathcal{O}$ and $\bm{t^{\prime}}|\bm{t}$ imply $\bm{t^{\prime}}\in\mathcal{O}$ . For an order ideal $\mathcal{O}\subset\mathbb{T}(\bm{x})$ , the corner set of $\mathcal{O}$ , denoted by $\mathcal{C}[\mathcal{O}]$ , is the set

\mathcal{C}[\mathcal{O}]=\{\bm{t}\in\mathbb{T}(\bm{x}):\bm{t}\notin\mathcal{O},x_{i}|\bm{t}\Rightarrow\bm{t}/x_{i}\in\mathcal{O},1\leq i\leq d\}.

Fix a monomial order $\prec$ on $\mathbb{T}(\bm{x})$ , for all $0\neq f\in\mathbb{F}[\bm{x}]$ , we may write

f=c_{{\bm{\gamma}}^{(1)}}\bm{x}^{{\bm{\gamma}}^{(1)}}+c_{{\bm{\gamma}}^{(2)}}\bm{x}^{{\bm{\gamma}}^{(2)}}+\cdots+c_{{\bm{\gamma}}^{(r)}}\bm{x}^{{\bm{\gamma}}^{(r)}}

where $0\neq c_{{\bm{\gamma}}^{(i)}}\in\mathbb{F},\bm{\gamma}^{(i)}\in\mathbb{N}^{d},i=1,\ldots,r$ , and $\bm{x}^{{\bm{\gamma}}^{(1)}}\succ\bm{x}^{{\bm{\gamma}}^{(2)}}\succ\cdots\succ\bm{x}^{{\bm{\gamma}}^{(r)}}$ . We shall call $\mathrm{LT}_{\prec}(f):=c_{{\bm{\gamma}}^{(1)}}\bm{x}^{{\bm{\gamma}}^{(1)}}$ the leading term and $\mathrm{LM}_{\prec}(f):=\bm{x}^{{\bm{\gamma}}^{(1)}}$ the leading monomial of $f$ .

Given an ideal $\mathcal{I}$ and a monomial order $\prec$ , there exists a unique reduced Gröbner basis $G_{\prec}$ for $\mathcal{I}$ w.r.t. $\prec$ . Suppose that $G_{\prec}=\{g_{1},\ldots,g_{m}\}$ , then the set

\mathcal{N}_{\prec}(\mathcal{I}):=\{\bm{x}^{\bm{\alpha}}\in\mathbb{T}(\bm{x}):\mathrm{LT}_{\prec}(g_{j})\nmid\bm{x}^{\bm{\alpha}},\hbox{~for~all~}1\leq j\leq m\}

is called the Gröbner éscalier of $\mathcal{I}$ w.r.t. $\prec$ . From the theory of Gröbner bases, we know that $\mathcal{N}_{\prec}(\mathcal{I})$ is an order ideal, and

\mathcal{C}[\mathcal{N}_{\prec}(\mathcal{I})]=\{\mathrm{LT}_{\prec}(g_{1}),\ldots,\mathrm{LT}_{\prec}(g_{m})\}.

If $P$ is a finite-rank ideal projector on $\mathbb{F}[\bm{x}]$ , then there are two important subsets of $\mathbb{F}[\bm{x}]$ associated with $P$ . The range of $P$ is defined as

V:=\mathrm{ran}P=\{p\in\mathbb{F}[\bm{x}]:p=Pf\mbox{~for~some~}f\in\mathbb{F}[\bm{x}]\},

which is a finite-dimensional subspace of $\mathbb{F}[\bm{x}]$ , and the kernel space of $P$

\mathrm{ker}P=\{g\in\mathbb{F}[\bm{x}]:Pg=0\},

which forms a zero-dimensional ideal in $\mathbb{F}[\bm{x}]$ . Furthermore, as an infinite-dimensional $\mathbb{F}$ -vector space, $\mathbb{F}[\bm{x}]$ has a corresponding dual space $(\mathbb{F}[\bm{x}])^{\prime}$ . An ideal projector $P$ on $\mathbb{F}[\bm{x}]$ also has a dual projector $P^{*}$ on $(\mathbb{F}[\bm{x}])^{\prime}$ , and the range of $P^{*}$ is

\Lambda:=\mathrm{ran}P^{*}={(\mathrm{ker}P)}^{\perp}=\{\lambda\in(\mathbb{F}[\bm{x}])^{\prime}:\mathrm{ker}P\subset\mathrm{ker}\lambda\}.

Indeed, $\Lambda$ is the set of interpolation conditions matched by $P$ . It’s easy to see that $\dim\Lambda=\dim V$ and

\mathrm{ker}\Lambda:=\{f\in\mathbb{F}[\bm{x}]:\lambda(f)=0\hbox{~for~all~}\lambda\in\Lambda\}=\mathrm{ker}P

which satisfies

\mathrm{ker}\Lambda\cap V=\{0\}.

The following theorems summarize some of the simple properties of ideal projectors.

Theorem 1.

([5]) A linear operator $P:\mathbb{F}[\bm{x}]\rightarrow\mathbb{F}[\bm{x}]$ is an ideal projector if and only if the equality

P(fg)=P(fPg)

holds for all $f,g\in\mathbb{F}[\bm{x}]$ .

Theorem 2.

([16, 17]) A linear operator $P:\mathbb{F}[\bm{x}]\rightarrow\mathbb{F}[\bm{x}]$ is an ideal projector if and only if the operator $P^{\prime}:=I-P$ satisfies

P^{\prime}(fg)=fP^{\prime}(g)+P^{\prime}(fPg)

for all $f,g\in\mathbb{F}[\bm{x}]$ .

3 Main results

In this section, we will describe a special class of ideal projectors with “good” error formulas in terms of ideal bases and interpolation conditions respectively.

3.1 Representation in terms of ideal bases

Following T. Sauer [10], we refer to a reduced Gröbner basis $G$ for an ideal $\mathcal{I}$ as a universal Gröbner basis if $G$ is a unique reduced Gröbner basis for $\mathcal{I}$ , independent of the monomial order. Now, we begin with an easy lemma about the universal Gröbner bases.

Lemma 3.

If the ideal $\mathrm{ker}P$ has a reduced Gröbner basis $G=\{g_{1},\ldots,g_{m}\}$ w.r.t. some monomial order, and the polynomials of $G$ have the form

g_{j}={\bm{x}}^{\bm{\alpha}^{(j)}}-\sum_{0\leq\bm{\beta}<{\bm{\alpha}^{(j)}}}c_{j,\bm{\beta}}\bm{x}^{\bm{\beta}},\quad\hbox{~for~}1\leq j\leq m\hbox{~~and~}c_{j,\bm{\beta}}\in\mathbb{F},

(1)

then $G$ is a universal reduced Gröbner basis for $\mathrm{ker}P$ w.r.t. any monomial order, and the monomial set

\mathcal{O}=\{{\bm{x}}^{\bm{\beta}}\in\mathbb{T}({\bm{x}}):{\bm{x}}^{\bm{\alpha}^{(j)}}\nmid{\bm{x}}^{\bm{\beta}},\hbox{~for~all~}1\leq j\leq m\}

(2)

is the unique Gröbner éscalier of $\mathrm{ker}P$ w.r.t. any monomial order.

Proof 1.

For an arbitrary $j$ with $1\leq j\leq m$ and an arbitrary ${\bm{\beta}}\in\mathbb{N}^{d}$ with $0\leq{\bm{\beta}}<{\bm{\alpha}}^{(j)}$ , we have that $\bm{x}^{\bm{\beta}}\mid{{\bm{x}}^{\bm{\alpha}^{(j)}}}$ . Suppose that $\prec$ is an arbitrary monomial order, then $\bm{x}^{\bm{\beta}}\mid{{\bm{x}}^{\bm{\alpha}^{(j)}}}$ together with $\bm{\beta}\neq{\bm{\alpha}^{(j)}}$ implies $\bm{x}^{\bm{\beta}}\prec{{\bm{x}}^{\bm{\alpha}^{(j)}}}$ . Consequently, for an arbitrary monomial order $\prec$ , $\mathrm{LT}_{\prec}(g_{j})={\bm{x}}^{\bm{\alpha}^{(j)}}$ with $1\leq j\leq m$ , and $S$ -polynomial of $g_{i}$ and $g_{j}$ with $1\leq i<j\leq m$ is the combination

S(g_{i},g_{j})=\frac{\mathrm{LCM}({\bm{x}}^{\bm{\alpha}^{(i)}},{\bm{x}}^{\bm{\alpha}^{(j)}})}{{\bm{x}}^{\bm{\alpha}^{(i)}}}g_{i}-\frac{\mathrm{LCM}({\bm{x}}^{\bm{\alpha}^{(i)}},{\bm{x}}^{\bm{\alpha}^{(j)}})}{{\bm{x}}^{\bm{\alpha}^{(j)}}}g_{j},

where $\mathrm{LCM}({\bm{x}}^{\bm{\alpha}^{(i)}},{\bm{x}}^{\bm{\alpha}^{(j)}})$ is the least common multiple of ${\bm{x}}^{\bm{\alpha}^{(i)}}$ and ${\bm{x}}^{\bm{\alpha}^{(j)}}$ .

Since $G$ is a reduced Gröbner basis for $\mathrm{ker}P$ w.r.t. some monomial order, it follows that $S(g_{i},g_{j})$ reduces to zero module $G$ w.r.t. this monomial order. Indeed, for arbitrary monomial order $\prec$ , $\mathrm{LT}_{\prec}(g_{j})={\bm{x}}^{\bm{\alpha}^{(j)}}$ , it implies that $S(g_{i},g_{j})$ reduces to zero module $G$ w.r.t. any monomial order. Therefore, we can say that $G$ is a universal reduced Gröbner basis for $\mathrm{ker}P$ w.r.t. any monomial order. Furthermore, it follows that $\mathcal{O}$ is the unique Gröbner éscalier of $\mathrm{ker}P$ w.r.t. any monomial order. ∎

Proposition 4.

Let $\{{g_{1}},\ldots,{g_{m}}\}$ be a reduced Gröbner basis for $\mathrm{ker}P$ satisfying condition (1), and $\mathcal{O}$ be a monomial set as in (2). Then for every monomial $\bm{x}^{\bm{\gamma}}\in\mathbb{T}(\bm{x})$ , there exist polynomials $A_{\bm{\gamma},j},1\leq j\leq m$ such that

P^{\prime}(\bm{x}^{\bm{\gamma}})=\bm{x}^{\bm{\gamma}}-P(\bm{x}^{\bm{\gamma}})=\sum_{j=1}^{m}A_{\bm{\gamma},j}g_{j}

(3)

and

A_{\bm{\gamma},j}=0\quad\mbox{if}\quad\bm{x}^{\bm{\alpha}^{(j)}}\nmid\bm{x}^{\bm{\gamma}}.

(4)

In other words,

P^{\prime}(\bm{x}^{\bm{\gamma}})=\bm{x}^{\bm{\gamma}}-P(\bm{x}^{\bm{\gamma}})=\sum_{\bm{\alpha}^{(j)}\leq\bm{\gamma}}A_{\bm{\gamma},j}g_{j}.

(5)

Proof 2.

For every $\bm{\gamma}\in\mathbb{N}^{d}$ , define an ideal

J_{\bm{\gamma}}=\langle g_{j}:\bm{\alpha}^{(j)}\leq\bm{\gamma}\rangle

To prove this proposition, it suffices to show that $\bm{x}^{\bm{\gamma}}-P(\bm{x}^{\bm{\gamma}})\in J_{\bm{\gamma}}$ for every $\bm{x}^{\bm{\gamma}}\in\mathbb{T}(\bm{x})$ . Assume not and let $\bm{x}^{\bm{\gamma}}$ be a monomial of least total degree such that ${\bm{x}}^{\bm{\gamma}}-P({\bm{x}}^{\bm{\gamma}})\not\in J_{\bm{\gamma}}$ . Since for every $\bm{x}^{\bm{\beta}}\in\mathcal{O}$ ,

0=\bm{x}^{\bm{\beta}}-P(\bm{x}^{\bm{\beta}})\in J_{\bm{\gamma}},

we know that $\bm{x}^{\bm{\gamma}}\not\in\mathcal{O}$ . Therefore, we can find some $1\leq j\leq m$ such that $\bm{\alpha}^{(j)}\leq\bm{\gamma}$ . Let $\bm{\delta}=\bm{\gamma}-\bm{\alpha}^{(j)}\geq\bm{0}$ . By Lemma 3, we have

\bm{x}^{\bm{\delta}}g_{j}=\bm{x}^{\bm{\gamma}}-\sum_{\bm{\beta}\in\mathcal{O},\bm{\beta}<{\bm{\alpha}}^{(j)}}c_{j,\bm{\beta}}\bm{x}^{\bm{\beta}+\bm{\delta}}.

Consequently,

\bm{x}^{\bm{\delta}}g_{j}=P^{\prime}(\bm{x}^{\bm{\delta}}g_{j})=P^{\prime}(\bm{x}^{\bm{\gamma}})-\sum_{\bm{\beta}\in\mathcal{O},\bm{\beta}<{\bm{\alpha}}^{(j)}}c_{j,\bm{\beta}}P^{\prime}(\bm{x}^{\bm{\beta}+\bm{\delta}})\in J_{\bm{\gamma}}.

But for every $\bm{\beta}$ such that $\bm{\beta}\in\mathcal{O},\bm{\beta}<{\bm{\alpha}}^{(j)}$ we have $\bm{\beta}+\bm{\delta}<\bm{\gamma}$ . Recall that $\bm{x}^{\bm{\gamma}}$ is a monomial of least total degree such that $P^{\prime}({\bm{x}}^{\bm{\gamma}})\not\in J_{\bm{\gamma}}$ . Hence, $P^{\prime}(\bm{x}^{\bm{\beta}+\bm{\delta}})\in J_{{\bm{\beta}+\bm{\delta}}}\subset J_{\bm{\gamma}}$ . Since $J_{\bm{\gamma}}$ is an ideal, then $P^{\prime}({\bm{x}}^{\bm{\gamma}})\in J_{\bm{\gamma}}$ . This is a contradiction to our hypothesis. ∎

We need a standard key lemma for factorization of homomorphisms.

Lemma 5.

([13, 16]) Let $A:X\rightarrow Y$ and $B:X\rightarrow Z$ be two linear operators between linear spaces $X,Y$ and $Z$ . Then there exists linear operator C such that

A=CB

if and only if

\mathrm{ker}B\subset\mathrm{ker}A.

The fact that an ideal projector $P$ has a “good” error formula depends on not only the ideal basis for $\mathrm{ker}P$ , but also the choice of $\mathrm{ran}P$ . Next is the main theorem of this paper, which states that the ideal projectors satisfying the conditions of Theorem 6 have “good” error formulas.

Theorem 6.

Suppose that an ideal $\mathrm{ker}P$ has a universal reduced Gröbner basis $G$ satisfying the conditions of Lemma 3, and $\mathrm{ran}P$ is

V=\mathrm{span}_{\mathbb{F}}\{{\bm{x}}^{\bm{\beta}}\in\mathbb{T}({\bm{x}}):{\bm{x}}^{\bm{\alpha}^{(j)}}\nmid{\bm{x}}^{\bm{\beta}},\hbox{~for~all~}1\leq j\leq m\}.

(6)

Then $G$ is the ideal basis for $\mathrm{ker}P$ that admits a “good” error formula.

Proof 3.

Define operators $A_{j}$ on $\mathbb{T}(\bm{x})$ by letting

A_{j}({\bm{x}}^{\bm{\gamma}})=\left\{\begin{array}[]{ll}A_{j,\bm{\gamma}},&\mbox{if}~{\bm{\alpha}}^{(j)}\leq\bm{\gamma};\\ 0,&\hbox{otherwise.}\end{array}\right.

(7)

where $A_{j,\bm{\gamma}}$ are defined in (5) and extend $A_{j,\bm{\gamma}}$ by linearity on $\mathbb{F}[\bm{x}]$ . Then by (5) and linearity, we have

f-Pf=P^{\prime}f=A_{j}(f)g_{j}.

By (7),

\mathrm{ker}A_{j}\supseteq\mathrm{span}\{{\bm{x}}^{\bm{\gamma}}:\bm{x}^{\bm{\alpha}^{(j)}}\nmid\bm{x}^{\bm{\gamma}}\}=\mathrm{ker}(\frac{1}{\bm{\alpha}^{(j)}!}D^{\bm{\alpha}^{(j)}}).

Hence, by Lemma 5, there exist operators $C_{j}$ such that $A_{j}=C_{j}\circ H_{j}(D)$ where $H_{j}(D):=\frac{1}{\bm{\alpha}^{(j)}!}D^{\bm{\alpha}^{(j)}}$ . It is trivial to check that $H_{j}(D)(g_{k})=\delta_{j,k}$ . ∎

In the following, we will present some examples to illustrate the conclusion of Theorem 6.

Example 1.

Let $P$ be a Lagrange projector onto $\mathrm{span}_{\mathbb{F}}\{1,x_{1},x_{2},x_{2}^{2}\}$ with the interpolation point set $\{(1,0),(1,1),(1,2),(2,0)\}\subset\mathbb{F}^{2}$ . Then by Theorem 6,

\{(x_{1}-1)(x_{1}-2),x_{2}(x_{2}-1)(x_{2}-2),x_{2}(x_{1}-1)\}

is the ideal basis for $\mathrm{ker}P$ that admits a “good” error formula. ∎

Example 2.

Let $P$ be an ideal projector onto $\mathrm{span}_{\mathbb{F}}\{1,x_{1},x_{2},x_{1}^{2},x_{1}x_{2},x_{2}^{2},x_{1}^{3}\}$ given by

	$\displaystyle Px_{1}^{2}x_{2}=0$
	$\displaystyle Px_{2}^{3}=x_{2}$
	$\displaystyle Px_{1}x_{2}^{2}=x_{1}x_{2}$
	$\displaystyle Px_{1}^{4}=2x_{1}^{3}-x_{1}^{2}.$

Then the ideal basis

\{x_{1}^{2}x_{2}-Px_{1}^{2}x_{2},x_{2}^{3}-Px_{2}^{3},x_{1}x_{2}^{2}-Px_{1}x_{2}^{2},x_{1}^{4}-Px_{1}^{4}\}

admits a “good” error formula. ∎

Example 3.

Let $P$ be a Lagrange projector onto the $\mathrm{span}_{\mathbb{F}}\{1,x_{1},x_{2},x_{3}\}$ with the interpolation point set $\{(0,0,0),(0,1,0),(0,0,1),(1,0,1)\}\subset\mathbb{F}^{3}$ . Then

\{x_{1}x_{2},x_{2}x_{3},x_{1}^{2}-x_{1},x_{2}^{2}-x_{2},x_{3}^{2}-x_{3},x_{1}x_{3}-x_{1}\}

is the ideal basis for $\mathrm{ker}P$ that admits a “good” error formula. ∎

We select test functions

	$\displaystyle f_{1}(x_{1},x_{2})$	$\displaystyle=(1-x_{1})^{2}+(1-x_{2})^{2}+1,$
	$\displaystyle f_{2}(x_{1},x_{2})$	$\displaystyle=x_{1}^{3}+x_{2}^{3},$
	$\displaystyle f_{3}(x_{1},x_{2},x_{3})$	$\displaystyle=(1-x_{1})^{2}+(1-x_{2})^{2}+(1-x_{3})^{2}+1,$
	$\displaystyle f_{4}(x_{1},x_{2},x_{3})$	$\displaystyle=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}$

to illustrate the “good” error formulas about the ideal projectors in the above examples.

For Example 1, we have

	$\displaystyle f_{1}-Pf_{1}$	$\displaystyle=(x_{1}-1)(x_{1}-2),$
	$\displaystyle f_{2}-Pf_{2}$	$\displaystyle=(x_{1}+3)(x_{1}-1)(x_{1}-2)+x_{2}(x_{2}-1)(x_{2}-2).$

For Example 2, we get

	$\displaystyle f_{1}-Pf_{1}$	$\displaystyle=0,$
	$\displaystyle f_{2}-Pf_{2}$	$\displaystyle=x_{2}^{3}-x_{2}.$

For Example 3,

	$\displaystyle f_{3}-Pf_{3}$	$\displaystyle=x_{1}^{2}-x_{1}+x_{2}^{2}-x_{2}+x_{3}^{2}-x_{3},$
	$\displaystyle f_{4}-Pf_{4}$	$\displaystyle=(x_{1}+1)(x_{1}^{2}-x_{1})+(x_{2}+1)(x_{2}^{2}-x_{2})+(x_{3}+1)(x_{3}^{2}-x_{3}).$

3.2 Representation in terms of interpolation conditions

Next, we will describe the properties of the interpolation conditions matched by the ideal projectors satisfying the conditions of Theorem 6.

Proposition 7.

Let $P$ be an ideal projector, and $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ the set of interpolation conditions matched by $P$ . Let $\prec_{lex(i)}$ , $1\leq i\leq d$ , be the lexicographic order

x_{i}\succ\cdots\succ x_{d}\succ x_{1}\succ\cdots\succ x_{i-1}.

Then $\mathrm{ker}P$ has a universal reduced Gröbner basis $G$ satisfying the conditions of Lemma 3 if and only if

\mathcal{N}_{\prec_{lex(1)}}(\mathrm{ker}\Lambda),\mathcal{N}_{\prec_{lex(2)}}(\mathrm{ker}\Lambda),\ldots,\mathcal{N}_{\prec_{lex(d)}}(\mathrm{ker}\Lambda)

are identical.

Proof 4.

One direction of the proof is obvious due to the fact that $\mathrm{ker}P$ has a universal reduced Gröbner basis $G$ satisfying the conditions of Lemma 3.

To prove the converse, assume that $G_{\prec_{lex(i)}}$ , $1\leq i\leq d$ , is the reduced Gröbner basis for $\mathrm{ker}\Lambda$ w.r.t. $\prec_{lex(i)}$ . Indeed, if

\mathcal{N}_{\prec_{lex(1)}}(\mathrm{ker}\Lambda)=\mathcal{N}_{\prec_{lex(2)}}(\mathrm{ker}\Lambda)=\cdots=\mathcal{N}_{\prec_{lex(d)}}(\mathrm{ker}\Lambda)=\mathcal{O},

then it is easy to prove

G_{\prec_{lex(1)}}=G_{\prec_{lex(2)}}=\cdots=G_{\prec_{lex(d)}}=G.

Suppose that $G=\{g_{1},\ldots,g_{m}\}$ and $\mathcal{C}[\mathcal{O}]=\{{\bm{x}}^{\bm{\alpha}^{(1)}},\ldots,{\bm{x}}^{\bm{\alpha}^{(m)}}\}$ . Then rearranging the elements of $\mathcal{C}[\mathcal{O}]$ appropriately, we have

\mathrm{LT}_{\prec_{lex(1)}}(g_{j})=\mathrm{LT}_{\prec_{lex(2)}}(g_{j})=\cdots=\mathrm{LT}_{\prec_{lex(d)}}(g_{j})={{\bm{x}}^{\bm{\alpha}^{(j)}}},\forall 1\leq j\leq m.

Since for arbitrary fixed $1\leq i\leq d$ , $G$ is the reduced Gröbner basis for $\mathrm{ker}\Lambda$ w.r.t. $\prec_{lex(i)}$ , it follows that the polynomials of $G$ have the form

g_{j}={{\bm{x}}^{\bm{\alpha}^{(j)}}}-\sum_{{\bm{x}}^{\bm{\beta}}{\prec_{lex(i)}}{{\bm{x}}^{\bm{\alpha}^{(j)}}}\atop{\bm{x}}^{\bm{\beta}}\in\mathcal{O}}c_{j,\bm{\beta}}\bm{x}^{\bm{\beta}},\quad\hbox{~for~all~}1\leq j\leq m.

Furthermore, by the property of lexicographic order, for arbitrary $1\leq i\leq d$ , ${\bm{x}}^{\bm{\beta}}{\prec_{lex(i)}}{{\bm{x}}^{\bm{\alpha}^{(j)}}}$ implies that $0\leq\bm{\beta}<{\bm{\alpha}}^{(j)}$ . Hence, we can deduce that the polynomials of $G$ have the form:

g_{j}={\bm{x}}^{\bm{\alpha}^{(j)}}-\sum_{0\leq\bm{\beta}<{\bm{\alpha}}^{(j)}}c_{j,\bm{\beta}}\bm{x}^{\bm{\beta}},\quad\hbox{~for~all~}1\leq j\leq m.

This completes the proof. ∎

Moreover, Proposition 7 coupled with Theorem 6 immediately implies the following useful corollary.

Corollary 8.

Let $P$ be an ideal projector, and $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ the set of interpolation conditions matched by $P$ . If

\mathcal{N}_{\prec_{lex(1)}}(\mathrm{ker}\Lambda)=\mathcal{N}_{\prec_{lex(2)}}(\mathrm{ker}\Lambda)=\cdots=\mathcal{N}_{\prec_{lex(d)}}(\mathrm{ker}\Lambda)=\mathcal{O},

(8)

where $\mathcal{N}_{\prec_{lex(i)}}(\mathrm{ker}\Lambda),1\leq i\leq d$ are as above, then the ideal projector $P$ onto

V=\mathrm{span}_{\mathbb{F}}\{\bm{x}^{\bm{\beta}}:\bm{x}^{\bm{\beta}}\in\mathcal{O}\}

(9)

has a “good” error formula.

Remark 1.

Let $\bm{\xi}^{(1)},\ldots,\bm{\xi}^{(\mu)}\in\mathbb{F}^{d}$ be distinct points and $\mathcal{A}^{(1)},\ldots,\mathcal{A}^{(\mu)}\subset\mathbb{N}^{d}$ lower sets. Suppose that the set of interpolation conditions has the form

\Lambda=\{\delta_{\bm{\xi}^{(k)}}\circ D^{\bm{\alpha}}:{\bm{\alpha}}\in\mathcal{A}^{(k)}~\mbox{for~all}~1\leq k\leq\mu\},

where $\delta_{\bm{\xi}^{(k)}}$ denotes the evaluation functional at the site $\bm{\xi}^{(k)}$ , then $\mathcal{N}_{\prec_{lex(i)}}(\mathrm{ker}\Lambda)$ can be directly computed by the fast algorithms given in [6, 8] without computing the Gröbner basis for $\mathrm{ker}\Lambda$ .

Example 4.

Suppose that the set of interpolation conditions matched by $P$ is as follows:

\Lambda=\{\delta_{(0,0)},\delta_{(0,0)}\circ D^{(0,1)},\delta_{(0,0)}\circ D^{(1,0)},\delta_{(0,1)},\delta_{(0,1)}\circ D^{(1,0)},\delta_{(1,0)},\delta_{(1,0)}\circ D^{(1,0)}\}.

Since

\mathcal{N}_{\prec_{lex(1)}}(\mathrm{ker}\Lambda)=\mathcal{N}_{\prec_{lex(2)}}(\mathrm{ker}\Lambda)=\{1,x_{2},x_{2}^{2},x_{1},x_{1}x_{2},x_{1}^{2},x_{1}^{3}\},

then $P$ onto $\mathrm{span}_{\mathbb{F}}\{1,x_{2},x_{2}^{2},x_{1},x_{1}x_{2},x_{1}^{2},x_{1}^{3}\}$ has a “good” error formula. ∎

As mentioned by Carl de Boor in [5], the existence of a “good” error formula for an ideal projector restricts the range of ideal projector to be of least degree. The following theorem is a particular case of this fact. Here, we also provide a simple proof, for completeness.

We denote by $\Pi_{r}$ the subspace of polynomials in $\mathbb{F}[\bm{x}]$ of total degree at most $r$ . Suppose $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}\subset(\mathbb{F}[\bm{x}])^{\prime}$ and $f\in\mathbb{F}[\bm{x}]$ , we write $\Lambda(f)={\left(\lambda_{1}f,\lambda_{2}f,\ldots,\lambda_{n}f\right)}^{T}$ . For a finite set $F=\{f_{1},\cdots,f_{k}\}\subset\mathbb{F}[\bm{x}]$ , $\Lambda(F)$ signifies the $n\times k$ matrix whose columns are $\Lambda(f_{i})$ , $1\leq i\leq k$ .

Theorem 9.

Let $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ be the set of interpolation conditions, If $\Lambda$ satisfies condition (8), then $V$ defined in (9) is the minimal degree interpolation space w.r.t. $\Lambda$ .

Proof 5.

Suppose that the maximal total degree of the monomials in $\mathcal{O}$ is $r$ , then $V\subset\Pi_{r}$ . It’s obvious that the interpolation problem of finding $p\in V$ such that

\lambda_{i}p=\lambda_{i}f,\quad\quad\hbox{for~all~}1\leq i\leq n

is poised. According to T. Sauer’s definition (cf. [9]), we need to prove two properties on this special class of projectors. Firstly, the operator $P$ onto $V$ is degree-reducing, namely for each $f\in\Pi_{k}$ with $0\leq k\leq r$ , the interpolating polynomial $Pf$ also belongs to $\Pi_{k}$ . Secondly, the subspace $V\subset\Pi_{r}$ is of minimal degree, namely there is no subspace $V^{\prime}\subset\Pi_{r-1}$ such that the above interpolation problem is poised.

Since each $f\in\Pi_{k}$ can be written in the form

f=\sum_{\bm{x}^{\bm{\beta}}\in\mathcal{O}\atop\bm{x}^{\bm{\beta}}\in\Pi_{k}}c_{\bm{\beta}}\bm{x}^{\bm{\beta}}+\sum_{\bm{x}^{\bm{\beta}}\not\in\mathcal{O}\atop\bm{x}^{\bm{\beta}}\in\Pi_{k}}c_{\bm{\beta}}\bm{x}^{\bm{\beta}},

then

Pf=\sum_{\bm{x}^{\bm{\beta}}\in\mathcal{O}\atop\bm{x}^{\bm{\beta}}\in\Pi_{k}}c_{\bm{\beta}}P(\bm{x}^{\bm{\beta}})+\sum_{\bm{x}^{\bm{\beta}}\not\in\mathcal{O}\atop\bm{x}^{\bm{\beta}}\in\Pi_{k}}c_{\bm{\beta}}P(\bm{x}^{\bm{\beta}}).

(10)

Since $V$ is the range of $P$ , we have that for any $\bm{x}^{\bm{\beta}}\in\mathcal{O}$ ,

P(\bm{x}^{\bm{\beta}})=\bm{x}^{\bm{\beta}}.

(11)

On the other hand, if $\bm{x}^{\bm{\beta}}\not\in\mathcal{O}$ , then there must exist some ${\bm{x}}^{\bm{\alpha}}\in\mathcal{C}[O]$ such that ${\bm{x}}^{\bm{\alpha}}|{\bm{x}}^{\bm{\beta}}$ . From Corollary 8, it follows that for some $c_{\bm{\gamma}}\in\mathbb{F}$ with $0\leq{\bm{\gamma}}<{\bm{\alpha}}$ ,

{\bm{x}}^{\bm{\alpha}}-\sum_{0\leq{\bm{\gamma}}<{\bm{\alpha}}}c_{\bm{\gamma}}\bm{x}^{\bm{\gamma}}\in\mathrm{ker}P.

Multiplying the above equation by ${\bm{x}}^{\bm{\beta}-\bm{\alpha}}$ , we get

{\bm{x}}^{\bm{\beta}}-\sum_{\bm{\beta}-\bm{\alpha}\leq{\bm{\gamma}+\bm{\beta}-\bm{\alpha}}<{\bm{\beta}}}c_{\bm{\gamma}}\bm{x}^{\bm{\gamma}+\bm{\beta}-\bm{\alpha}}\in\mathrm{ker}P.

If $\bm{x}^{\bm{\gamma}+\bm{\beta}-\bm{\alpha}}\not\in\mathcal{O}$ , we repeat the above processing. Finally, we can find some $\bm{x}^{\bm{\beta}^{\prime}}\in\mathcal{O}$ with $\bm{\beta}^{\prime}<\bm{\beta}$ , and associated coefficients $c_{\bm{\beta}^{\prime}}\in\mathbb{F}$ such that

{\bm{x}}^{\bm{\beta}}-\sum_{\bm{x}^{\bm{\beta}^{\prime}}\in\mathcal{O}\atop\bm{\beta}^{\prime}<\bm{\beta}}c_{\bm{\beta}^{\prime}}\bm{x}^{\bm{\beta}^{\prime}}\in\mathrm{ker}P.

Since $\bm{x}^{\bm{\beta}}\in\Pi_{k}$ , it follows that

P(\bm{x}^{\bm{\beta}})=\sum_{\bm{x}^{\bm{\beta}^{\prime}}\in\mathcal{O}\atop\bm{\beta}^{\prime}<\bm{\beta}}c_{\bm{\beta}^{\prime}}\bm{x}^{\bm{\beta}^{\prime}}\in\Pi_{k}.

(12)

From (10), (11), (12), we can conclude that for arbitrary $f\in\Pi_{k}$ with $1\leq k\leq r$ , $Pf\in\Pi_{k}$ .

To prove the minimal degree property, we set $\mathcal{O}^{\prime}=\mathcal{O}_{1}^{\prime}\bigcup\mathcal{O}_{2}^{\prime}$ , where

\mathcal{O}_{1}^{\prime}:=\{{\bm{x}}^{\bm{\beta}}\in\mathbb{T}(\bm{x}):{\bm{x}}^{\bm{\beta}}\in\Pi_{r-1}\hbox{~and~}{\bm{x}}^{\bm{\beta}}\in\mathcal{O}\},

and

\mathcal{O}_{2}^{\prime}:=\{{\bm{x}}^{\bm{\beta}}\in\mathcal{\mathbb{missing}}{T}(\bm{x}):{\bm{x}}^{\bm{\beta}}\in\Pi_{r-1}\hbox{~and~}{\bm{x}}^{\bm{\beta}}\not\in\mathcal{O}\}.

Then we need only to prove that the matrix $\Lambda(\mathcal{O}^{\prime})$ has rank less than $n$ .

Recalling equality (12), we can easily see that for an arbitrary ${\bm{x}}^{\bm{\beta}}\in\mathcal{O}_{2}^{\prime}$ , $\Lambda({\bm{x}}^{\bm{\beta}})$ linearly depends on the columns of $\Lambda(\mathcal{O}_{1}^{\prime})$ . Equivalently, $\Lambda(\mathcal{O}^{\prime})$ has rank less than or equal than $\#\mathcal{O}_{1}^{\prime}$ . Since at least one ${\bm{x}}^{\bm{\beta}}\in\mathcal{O}$ belongs to $\Pi_{r}$ and not to $\Pi_{r-1}$ , we have that the matrix $\Lambda(\mathcal{O}^{\prime})$ has rank less than $n$ . To sum up, we can say that $V=\mathrm{span}_{\mathbb{F}}\{\bm{x}^{\bm{\beta}}:\bm{x}^{\bm{\beta}}\in\mathcal{O}\}$ is the minimal degree interpolation space w.r.t. $\Lambda$ .∎

Acknowledgements

The authors wish to thank the anonymous reviewer for valuable suggestions and comments that have improved the presentation of the paper.

References

[1] T. Becker, V. Weispfenning, Gröbner Bases, Vol. 141 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1993.
[2] G. Birkhoff, The algebra of multivariate interpolation, in: C. V. Coffman, G. J. Fix (Eds.), Constructive Approaches to Mathematical Models, Academic Press, New York, 1979, pp. 345–363.
[3] C. de Boor, On the error in multivariate polynomial interpolation, Appl. Numer. Math. 10 (1992) 297–305.
[4] C. de Boor, The error in polynomial tensor-product, and Chung-Yao, interpolation, in: A. LeMéhauté, C. Rabut, L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt University Press, Nashville TN, 1997, pp. 35–50.
[5] C. de Boor, Ideal interpolation, in: C. K. Chui, M. Neamtu, L. L. Schumaker (Eds.), Approximation Theory XI: Gatlinburg 2004, Nashboro Press, Brentwood TN, 2005, pp. 59–91.
[6] L. Cerlienco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Disc. Math. 139 (1-3) (1995) 73–87.
[7] D. Cox, J. Little, D. O’Shea, Ideal, Varieties, and Algorithms, third ed., Undergraduate Texts in Mathematics, Springer, New York, 2007.
[8] B. Felszeghy, B. Ráth, L. Rónyai, The lex game and some applications, J. Symb. Comput. 41 (6) (2006) 663–681.
[9] T. Sauer, Polynomial interpolation of minimal degree, Numer. Math. 78 (1)(1997) 59–85.
[10] T. Sauer, Lagrange interpolation on subgrids of tensor product grids, Math. Comp. 73 (245) (2004) 181–190.
[11] T. Sauer, Y. Xu, On multivariate Lagrange interpolation, Math. Comp. 64 (1995) 1147–1170.
[12] T. Sauer, Y. Xu, On multivariate Hermite interpolation, Adv. Comput. Math. 4(1995) 207–259.
[13] B. Shekhtman, On one question of Ed Saff, Elec. Trans. Numer. Anal. 25 (2006) 439–445.
[14] B. Shekhtman, Ideal interpolation: Translations to and from Algebraic Geometry, in: L. Robbiano, J. Abbott (Eds.), Approximate Commutative Algebra, Texts and Monographs in Symbolic Computation, Springer-Vienna, New York, 2009, pp. 163–192.
[15] B. Shekhtman, On non-existence of certain error formulas for ideal interpolation, J. Approx. Theory 162 (7) (2010) 1398–1406.
[16] B. Shekhtman, On the naïve error formula for bivariate linear interpolation, in: Wavelets and Splines: Athens 2005, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2006, pp. 416–427.
[17] B. Shekhtman, On error formulas for multivariate polynomial interpolation, in: M. Neamtu, L. Schumaker (Eds.), Approximation Theory XII: San Antonio 2007, Nashboro Press, Brentwood TN, 2008, pp. 386–397.
[18] S. Waldron, The error in linear interpolation at the vertices of a simplex, SIAM J. Numer. Anal. 35 (3) (1998) 1191–1200.