Cubature Formulae and Polynomial Ideals

Advances in Applied Mathematics 23, 211᎐233 Ž1999. Article ID aama.1999.0652, available online at http:rrwww.idealibrary.com on

Cubature Formulae and Polynomial Ideals Yuan Xu1 Department of Mathematics, Uni¨ ersity of Oregon, Eugene, Oregon 97403-1222 E-mail: [email protected] Received July 1998; accepted September 19, 1998

The structure of cubature formulae of degree 2 n y 1 is studied from a polynomial ideal point of view. The main result states that if I is a polynomial ideal generated by a proper set of Ž2 n y 1.-orthogonal polynomials and if the cardinality of the variety V Ž I . is equal to the codimension of I, then there exists a cubature formula of degree 2 n y 1 based on the points in the variety. The result covers a number of cubature formulae in the literature, including Gaussian cubature formulae on one end and the usual product formulae on the classical domains on the other end. The result also offers a new method for constructing cubature formulae. 䊚 1999 Academic Press Key Words: Cubature formula, polynomial ideal, variety, orthogonal polynomials in several variables, common zeros.

1. INTRODUCTION The purpose of this paper is to study the structure of cubature formulae using the notion of polynomial ideal and its variety. Let Ł d s ⺢w x 1 , . . . , x d x be the space of polynomials in d real variables, and let Ł nd be the space of polynomials of degree at most n. It is known that dim Ł nd s n qd d . Let L be a square positive linear functional defined on Ł d, such as those given by L Ž f . s H⺢ d f Žx.W Žx. dx, where W is a nonnegative weight function with finite moments of all order. A cubature formula of degree 2 n y 1 with respect to L is a linear functional

ž /

N

Ln Ž f . [

Ý ␭k f Ž x k . ,

x k g ⺢ d,

␭ k g ⺢,

Ž 1.1.

ks1

where the points x 1 , . . . , x N are assumed to be distinct, such that LnŽ f . s 1

Supported by the National Science Foundation under Grant DMS-9802265. 211 0196-8858r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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L Ž f . for all f g Ł 2d ny1 and there exists at least one polynomial f U g Ł 2d n such that LnŽ f U . / L Ž f U .. The points x k are called nodes and ␭ k are called weights of the cubature formula. For a cubature formula we denote the set of its nodes by V, and denote the polynomial ideal which has V as its variety by I; that is, I s I Ž V . s  p g Ł d N pŽx. s 0, x g V 4 . We shall call I the generating ideal of the cubature formula. In terms of common zeros of orthogonal polynomials, the notion of variety and its relation to cubature formulae can be traced back to the work of Radon w9x, and were developed in the work of Žcf. w4x., among many others. For d s 1, a Mysovskikh Žcf. w8x. and Moller ¨ Gaussian quadrature formula of degree 2 n y 1, which uses only n nodes, exists if and only if its nodes are zeros of orthogonal polynomials of degree n. This elegant characterization was extended to functions of several variables by Mysovskikh using the concept of common zeros of orthogonal polynomials. Let us denote by Vn d the subspace of polynomials of degree d n that are orthogonal to all polynomials in Ł ny 1 with respect to the inner product defined by ² f, g : s L Ž fg .. It follows from the Gram᎐Schmidt process that dim Vn d s rnd, where rnd s n q nd y 1 is the dimension of the subspace of homogeneous polynomials of degree n. We denote by  Pkn4 an orthonormal basis of Vn d, where the superscript n means that Pkn is of degree n and the subindex k satisfies 1 F k F rnd. It is known that for a cubature formula of degree 2 n y 1 to exist, it is necessary that N G d Ž w x. dim Ł ny 1 cf. 10 . The cubature formulae that attain this lower bound are called Gaussian cubature. In w6x Mysovskikh proved that a Gaussian d cubature formula exists if and only if its N s dim Ł ny 1 nodes are comn mon zeros of ⺠n , where we use the notation ⺠n s Ž P1 , . . . , Prnnd ., which we interpret as either a set or a column vector. In other words, the ideal d generated by ⺠n has a variety of cardinality dim Ł ny1 . However, for d ) 1, Gaussian cubature formulae do not exist in general. Moller in w3x proved ¨ an improved lower bound for the number of nodes of a cubature formula of degree 2 n y 1, which shows, as a special case, that N G dim Ł 2ny 1 q w nr2x for d s 2 and L being a centrally symmetric linear functional. Moreover, he showed that the nodes of a cubature formula that attains this new lower bound are common zeros of a proper subset of ⺠n . Moller ¨ used the theory of polynomial ideal in his work, and introduced the concept such as H-basis and s-orthogonal to the subject. However, cubature formulae that attain the new lower bound also do not exist in general. To step beyond, we studied the structure of cubature formulae based on common zeros of a set of orthogonal and quasi-orthogonal polynomials recently, making use of a vector᎐matrix notion of orthogonal polynomials in several variables w12, 13x, where the structure of the generating ideal was analyzed although the term was not used. In these studies, the emphasis is

ž

/

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more or less on finding cubature formulae with minimal or close to minimal number of nodes. The present work stemmed from the realization that several results in w12x can be stated in terms of the generating ideal and its variety. It turns out that using the theory of polynomial ideal and variety, we can prove a general result which extends, and possibly sheds new light on, the results mentioned above. Indeed, we will show that if a cubature formula of degree 2 n y 1 exists, then the codimension of I is equal to the cardinality of V. More importantly, we will prove that if a polynomial ideal I is generated by a proper subset set of orthogonal polynomials Žsee Section 2. and its codimension is finite and equal to the cardinality of its variety, then a cubature formula of degree 2 n y 1 exists. This result extends several existing results in the literature, including those mentioned above, it also covers as an extreme case the classical product formulae on the standard domains. In particular, it is not restricted to minimal or near minimal cubature formulae. Moreover, starting from a subset of orthogonal polynomials that has a large number of common zeros, the result also offers a method of constructing cubature formulae. We will also give a necessary condition for a subset of orthogonal polynomials to have a large number of zeros. Several results in the paper are originated from w12x, but both their statements and their proof will be given anew from the ideal and variety point of view. The paper is organized as follows. In Section 2 we prove the main results and discuss their implication and relation to the previous results in the literature. In Section 3 we discuss various examples that illustrate our results.

2. IDEALS AND CUBATURE FORMULAE Let I be a polynomial ideal and V s V Ž I . be its affine variety. We consider the case where V is a zero-dimensional variety in ⺢ d, that is, V is a finite set of distinct points in ⺢ d. We denote by < V < the cardinality of V, which is the number of distinct elements in V. The codimension of I is denoted by codim I, that is, codim I s dim Ł drI. It is known that < V < F codim I ; see, for example, w1, p. 232, Proposition 8x, where the result is stated for ⺓w x 1 , . . . , x d x, but the proof works for any field. If I is generated by the

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polynomials f 1 , f 2 , . . . , f M , that is, every f g I can be written as M

f Ž x. s

Ý a j Ž x. f j Ž x. ,

aj g Ł d ,

js1

then we write I as I s ² f 1 , f 2 , . . . , f M : and we call f 1 , . . . , f M a basis of the ideal I. According to the Hilbert basis theorem, every polynomial ideal has a finite basis. If for any f g I the degree of a j f j is not greater than the degree of f in the above representation, then we call the basis an H-basis, a concept that goes back to Macauley and was used by Moller ¨ for the study of cubature formulae Žw3x.. For a finite set V in ⺢ d, we also consider the ideal I Ž V . associated to V, that is, I Ž V . s  p g Ł d < p Ž x . s 0,

x g V 4.

The ideal I Ž V . is the largest ideal having V as variety. We know that < V < s codim I Ž V .. If V is the set of nodes of a cubature formula, we call V the generating variety and I Ž V . the generating ideal of the cubature. To understand the structure of a cubature formula, we need the notions of orthogonal and quasi-orthogonal polynomials in several variables. We write the monomials x ␣ s x 1␣ 1 ⭈⭈⭈ x d␣ d for x g ⺢ d and ␣ s Ž ␣ 1 , . . . , ␣ d . g ⺞ d and use the usual multi-index notation. The polynomial x ␣ is of Žtotal. degree < ␣ < s ␣ 1 q ⭈⭈⭈ q␣ d . To order the monomials in Ł d s ⺢w x 1 , . . . , x d x, we use either graded lexicographic order or the graded reverse lexicographic order Žsee, for example, w1, Chap. 2x.. For our purpose, we assume that one particular order is chosen and fixed throughout this paper. Let L be a square positive linear functional defined on Ł d, that is, L Ž P 2 . ) 0 unless P s 0; examples include L defined by L Ž f . s Hf Žx.W Žx. dx. With the monomial order fixed, we can use the Gram᎐Schmidt process to get a sequence of orthonormal polynomials with respect to L . As in the introduction, we denote these orthonormal polynomials by Pkn, where 1 F k F rnd s n q nd y 1 , and the superscript n means that Pkn g Ł nd . We also use the notation ⺠n s Ž P1n, . . . , Pr nd . and regard it either as a set or as a column vector. One consequence of regarding ⺠n as a vector is the following three-term relation in vector-matrix form,

ž

/

x i Pn s A n , i ⺠nq1 q Bn , i ⺠n q ATny1, i ⺠ny1 ,

1 F i F d, n G 0, Ž 2.1.

where ⺠0 s 1 and ⺠y1 s 0, A n, i and Bn, i are matrices of proper size. This relation plays an important role in the general theory of orthogonal polynomials in several variables. Together with a rank condition on A n, i , the relation characterizes the orthogonality of orthogonal polynomials Žwe refer to survey w13x and the references there.. A polynomial Q sn is called a

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quasi-orthogonal polynomial of degree n and order s, if it is orthogonal to all polynomials of degree n y s y 1; that is, L Ž Q sn P . s 0,

d P g Ł nysy1 .

From the orthogonality of Q sn, we can write Q sn in terms of orthogonal polynomials of degree n y s up to n. Taking Pn as a column vector, each orthogonal polynomial of degree n can be written as aT ⺠n , where a is a scalar column vector. Hence, we can write Q sn as Q sn s aT0 ⺠n q ⭈⭈⭈ qaTs ⺠nys , where a i are scalar vectors of proper sizes. In particular, a quasi-orthogonal polynomial of degree n and order 0 is just an ordinary orthogonal polynomial. Assume a cubature formula of degree 2 n y 1 exists, we start with analyzing its generating ideal. Let x 1 , . . . , x N denote the distinct nodes of the cubature formula, N s < V <. Let us define ⌿nq k s Ž x ␣j . 1FjFN , < ␣ < 1Fnqk

and

␶ k s rank ⌿nqk ,

d where for each k G 0, ⌿nq k is a matrix of the size N = dim Ł nqk . Let us d further introduce the notation ␶ 0 s dim Ł ny1 q ␴ 0 and ␴r s ␶r y ␶ry1 for r ) 0. Our first result gives necessary conditions for the generating ideal and the generating variety of a cubature formula of degree 2 n y 1.

THEOREM 2.1. Let I be a generating ideal of a cubature formula of degree 2 n y 1. Define ⌿nq k as abo¨ e. If ␴mq1 s 0 for an m - n y 1, then a basis for I contains rnd y ␴ 0 linearly independent orthogonal polynomials of degree d n, rnq k y ␴ k linearly independent quasi-orthogonal polynomials of degree n q k and order 2 k for each 0 - k F m q 1. Moreo¨ er, ␴ k G 0 and d < V < s codim I s dim Ł ny 1 q ␴ 0 q ⭈⭈⭈ q␴m .

Proof. For each k we consider the linear system of equations

Ý

< ␣ < 1Fnqk

c␣ x ␣j s 0,

1 F j F N,

Ž Ek .

whose coefficient matrix is ⌿nq k . Each solution of the system Ž Ek . yields a polynomial of degree at most n q k that vanishes on all nodes of the d cubature formula, and there are exactly rnq k y ␶ k linearly independent d polynomials of this type. Since N G dim Ł ny 1 implies that there is no polynomial of degree n y 1 that will vanish on all nodes of the cubature

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formula, it follows that the matrix ⌿ny 1 has full rank; hence, ␶ 0 s d dim Ł ny 1 q ␴ 0 shows that ␴ 0 G 0. That ␴ k G 0 for k ) 0 follows from the fact that ␶ k is nondecreasing by definition. The linear system of equations Ž E0 . has exactly dim Ł nd y ␶ 0 linearly independent solutions, which leads to rnd y ␴ 0 polynomials of degree exactly n that vanish on all nodes. If P is one of these polynomials and Q is a polynomial of degree n y 1 or less, then using the fact that the cubature formula is of degree 2 n y 1 we conclude that N

L Ž PQ . s

Ý ␭ k P Ž x k . Q Ž x k . s 0, ks1

which shows that P is an orthogonal polynomial. For k ) 0, the system d Ž Ek . has exactly dim Ł nqk y ␶ k linearly independent solutions, which d leads to dim Ł nq k y ␶ k polynomials of degree at most n q k that vanish d on all nodes. Among these polynomials, exactly dim Ł nq ky1 y ␶ ky1 of them are polynomials of degree at most n q k y 1 that come from the solutions of Ž Enq ky1 .. Therefore, there are exactly d d d dim Ł nq k y ␶ k y Ž dim Ł nqky1 y ␶ ky1 . s r k y ␴ k

polynomials of degree n q k vanishing on all nodes of the cubature formula. Since the cubature formula is of degree 2 n y 1, it follows that these polynomials are orthogonal to all polynomials of degree Ž2 n y 1. y Ž n q k . s Ž n q k . y 2 k y 1; that is, they are quasi-orthogonal polynomid als of degree n q k and order 2 k. Since ␴mq 1 s 0, there are rnqmq1 many linearly independent quasi-orthogonal polynomials of degree n q m q 1 d Ž and order 2 m q 2 vanishing on V. It follows that codim I s dim Ł nq mr I d . l Ł nq m , which shows that all polynomials corresponding to the solutions of the systems of equations Ž Ek . for k G n q m q 1 belong to the ideal. Since each polynomial of degree k vanishing on V belongs necessarily to d .s the kernel of the system of equations Ž Ek ., we have dimŽ I l Ł nqm d d rnq m y ␶m . Hence, we conclude that codim I s ␶m s dim Ł ny1 q ␴ 0 q ⭈⭈⭈ q␴m . Using the notation ⺠n as a column vector, each orthogonal polynomial of degree n in the theorem can be written as a linear combination of Pin, or aT ⺠n for a scalar vector a. Instead of writing down the rnd y ␴ 0 linearly independent orthogonal polynomials in the theorem individually, we consider them as the components of the vector U0T ⺠n , where U0 is a matrix of the size Ž rnd y ␴ 0 . = rnd and U0 has full rank. In the following we may regard U0T ⺠n as a set as well. Likewise, we introduce the vector notation

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⺡ nq k, 2 k for a sequence of quasi-orthogonal polynomials of degree n q k of order 2 k, ⺡ nq k , 2 k s ⺠nqk q ⌫1⺠nqky1 q ⭈⭈⭈ q⌫2 k ⺠nyk , where ⌫i are matrices of proper size, we write the quasi-orthogonal polynomials of degree n q k and order 2 k in the theorem as UkT ⺡ nqk , d d d where Uk is a matrix of size Ž rnqk y ␴ k . = rnqk which has rank rnqk y ␴k . In the following, we will also write ⺡ n, 0 s ⺠n . We note that one important relation is T x i : span  UkT ⺡ nqk , 2 k 4 ¬ span  Ukq1 ⺡ nqkq1, 2 kq2 4 ,

Ž 2.2.

where x i is the ith coordinate of x and x i f means multiply f by x i . Indeed, using the three-term relation Ž2.1., we see that the polynomials in x i UkT ⺡ nqk, 2 k are quasi-orthogonal polynomials of degree n q k q 1 and order 2 k q 2. Let us also mention the notion of s-orthogonal introduced by Moller. A ¨ polynomial f is said to be s-orthogonal with respect to L , if L Ž fg . s 0 for any polynomial g such that fg g Ł ds . All polynomials in ⺡ nqk, 2 k for 0 F k F n y 1 are Ž2 n y 1.-orthogonal. With these notations, we can rewrite the first part of Theorem 2.1 as follows. COROLLARY 2.2. Let I be a generating ideal of a cubature formula of degree 2 n y 1. If ␴mq 1 s 0 for an m - n y 1, then I s ²U0T ⺠n , U1T ⺡ nq1, 2 , . . . , UmT ⺡ nqm , 2 m , ⺡ nqmq1, 2Ž mq1. : ,

Ž 2.3.

d where Uk has rank rnqk y ␴ k as abo¨ e. In particular, I is generated by a set of Ž2 n y 1.-orthogonal polynomials.

We note that the basis given in Ž2.3. may not be minimal, that is, it may contain more elements than what is necessary. For example, the polynomials in x i U0T ⺠n evidently belong to the ideal I, and by the relation Ž2.2. these polynomials are in spanU1T ⺡ nq1, 2 4 . Hence, part of U1T ⺡ nq1, 2 is redundant. In fact, the basis Ž2.3. is maximal in the sense that we actually have d T T T I l Ł nq mq1 s span  U0 ⺠n , U1 ⺡ nq1, 2 , . . . , Um ⺡ nqm , 2 m , ⺡ nqmq1, 2Ž mq1. 4 .

Ž 2.4. We include ⺡ nq mq1, 2Ž mq1. in the basis because our assumption ␴mq1 s 0 d d Ž . implies that Ł drI s Ł nq m r I l Ł nqm .

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The above theorem gives the necessary conditions satisfied by the generating ideal and variety of a cubature formulae of degree 2 n y 1. In the following we show that an ideal satisfying conditions as such will generate a cubature formula of degree 2 n y 1, which is our main result. We state the result in two equivalent forms, the first one uses the language of ideal and variety, the second does not. THEOREM 2.3. Let L be a square positi¨ e linear functional. Let I be a polynomial ideal generated by a set of Ž2 n y 1.-orthogonal polynomials as in Ž2.3. such that Ž2.4. satisfies, and let V be the ¨ ariety of I. Assume that d d Ž . < < Ł drI s Ł nq m r I l Ł nqm for some m F n y 1. If V s codim I, then there is a cubature formulae of degree 2 n y 1 whose nodes are the points in V. THEOREM 2.3X . Let L be a square positi¨ e linear functional. Let A be a set of polynomials, A s U0T ⺠n , U1T ⺡ nq1, 2 , . . . , UkT ⺡ nqk, 2 k 4 , where Uj are matrices, and denote by < V < the number of real common zeros of polynomials in A. Define the subspaces, U0 [ span U0T ⺠n , Uj [ span  x i Ujy1 , 1 F i F d, Uj T ⺡ nqj, 2 j 4 , 1 F j F n y 1. d If there is an m F n y 1 such that dim Um s rnqm and d d d < V < s dim Ł ny 1 q Ž r n y dim U0 . q ⭈⭈⭈ q Ž r nqm y dim Um . , Ž 2.5 .

then there is a cubature formulae of degree 2 n y 1 whose nodes are the common zeros of polynomials in A. The conditions of Theorem 2.3X look more complicated, but they are easy to check for applications; see the discussion after the proof of Theorem 2.3. To show that these two theorems are equivalent, we let I s ² A :. We only have to show that Ž2.5. is equivalent to codim I s < V <. Since Uj ; I for each j, it follows that the codimension of I is less than or equal to the right hand side of Ž2.5. in general. Hence, the fact that < V < F codim I shows that Ž2.5. implies codim I s < V <, which in turn shows that I l Ł nq mq1 s D j Uj . The other direction follows easily. One may compare Theorem 2.3 with a theorem of Moller on general cubature ¨ formulae; see w3, 7x. Proof of Theorem 2.3. Let I s ² A :, where A is as in Theorem 2.3X . d d Ž . Since Ł drI s Ł nq m r I l Ł nqm , we can apply the operation of multiplication by x i and use the relation Ž2.2. to enlarge the generating basis of I. Repeating the process, we end up with an enlarged basis in the form of Ž2.3., where all matrices Uj have full rank, such that Ž2.4. holds. We note that U1 , . . . , Uk may be different from what they are in A, since additional

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linearly independent elements could be introduced from the enlargement; we keep the notation for convenience. Let Vk be a matrix of size d Ž < . rnq k = ␴ k with rank ␴ k such that the matrix Uk Vk , whose columns are columns of Uk and Vk , is invertible. Define the space d Vn Ž I . s Ł ny1 j span  V0T ⺠n , V1T ⺡ nq1, 2 , . . . , VmT ⺡ nqm , 2 m 4 ,

which is the complement of the space spanned by the enlarged basis. Since d  4 Ž .  4 I l Ł ny 1 s 0 , we have V n I l I s 0 by its definition. Hence, we have d d d d Ž . Ž . ., it that Ł nq m s Vn I [ I l Ł nqm . Since Ł drI s Ł nqm rŽ I l Ł nqm Ž . follows that dim Vn I s codim I. We now prove that for any function f defined on V, there is a unique polynomial Pf in VnŽ I . such that Pf s f on V. Taking VkT ⺡ nqk, 2 k as a column vector, each polynomial P in VnŽ I . can be written as P Ž x. s

Ý

< ␣
m

c␣ x ␣ q

Ý c Tk VkT ⺡ nqk , 2 k , ks0

where we write ⺡ n, 0 s ⺠n and c k are scalar column vectors of proper sizes. Let x 1 , . . . , x N , N s < V <, denote the distinct points in V. Then Pf Žx i . s f Žx i . leads to a linear system of equations whose coefficient matrix is M s w ⌽ Žx 1 .< ⭈⭈⭈ < ⌽ Žx N .x, where ⌽ Žx. is a column vector defined by T

T

T T T T ⌽ Ž x . s X ny 1 , Ž V0 ⺠n Ž x . . , Ž V1 ⺡ nq1, 2 Ž x . . , . . . , Ž Vm ⺡ nqm , 2 m Ž x . .

T T

,

and X ny 1 denote the column vector X ny1 s wx ␣ x < ␣ < F ny1 in which the monomials are arranged according to our fixed monomial order. The matrix M is square, since < V < s codim I s dim VnŽ I .. Hence, it is sufficient to prove that M is invertible. Suppose it is not, then M has rank r - N. We may assume that the first r y 1 columns of the matrix are linearly independent and there exist scalars a1 , a2 , . . . , a r , not all zero, such that Ý rks 1 a k⌽ Ž x k . s 0. Moreover, we can assume that a1 and a r are not zero. In terms of the components of the vector ⌽, this shows that r

⌳Ž P . [

Ý ak P Ž x k . s 0,

d P g Ł ny1

and

ks1

P g VkT ⺡ nqk , 2 k , 0 F k F m. On the other hand, we clearly have ⌳Ž P . s 0 for any P g I, in particular, d for P g UkT ⺡ nqk, 2 k . From ⌳Ž P . s 0, P g Ł ny1 , and the fact that ŽU0 < V0 . is invertible, it follows that ⌳Ž P . s 0 for all P g Ł nd . Similarly, working

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YUAN XU

up through k s 1, 2 . . . and using the fact that ŽUk < Vk . are invertible, we d d can conclude that ⌳Ž P . s 0 for P g Ł nq m and, thus, for P g Ł since d ␣ d d Ł rI s Ł nq m rŽ I l Ł nqm .. In particular, it follows that ⌳Ž x i x . s 0 for any ␣ , where x i are coordinates of x. Writing x k s Ž x k, 1 , . . . , x k, d ., we conclude that r

xr , i

r

Ý

ak P Ž x k . y

ks1

Ý ak x k , i P Ž x k . s x r , i ⌳ Ž P . y ⌳ Ž x i P . s 0, ks1

Ž . Ž . for any P g V nŽ I .. That is, Ý ry1 ks1 a k x k, i y x r, i ⌽ x k s 0. Since ⌽ Žx 1 ., . . . , ⌽ Žx ry1 . are linearly independent, we conclude that a k Ž x k, i y x r, i . s 0 for 1 F k F r y 1 and 1 F i F d. Since a1 / 0, it follows that x 1, i s x r, i for 1 F i F d, that is, x 1 s x d , which contradicts the fact that x k are distinct. By the uniqueness of the polynomial Pf g VnŽ I . that satisfies Pf Žx k . s f Žx k ., we can write Pf as N

Pf Ž x . s

Ý f Ž x k . l k Ž x. ,

N s < V <,

ks1

where l k are unique polynomials in VnŽ I . determined by l k Žx j . s ␦ k, j , 1 F k, j F N. Applying the linear functional L on Pf leads to a cubature formula N

Ln Ž f . s

Ý ␭k f Ž x k .

where ␭ k s L Ž l k . ,

ks1

which satisfies, by the uniqueness of Pf , that LnŽ f . s L Ž f . for f g VnŽ I .. We need only to prove that the same equality holds for all f g Ł 2d ny1. For any polynomial f g I, we have LnŽ f . s 0 since f vanishes on the variety of I. On the other hand, by assumption, since I is generated by a basis of the form Ž2.3., it follows from the orthogonality that L Ž P . s 0 for each element P in the basis. Therefore, L Ž P . s 0 s LnŽ P . for P in the basis. Moreover, for each polynomial f g Ł nd , we can use the fact that ŽU0 < V0 . is d invertible to write f s g q h, where g g spanU0T ⺠n4 and h g Ł ny1 l T  4 Ž . span V0 ⺠n ; Vn I . Hence, it follows that Ln Ž f . s Ln Ž g . q Ln Ž h . s 0 q Ln Ž h . s 0 q L Ž h . s L Ž g . q L Ž h. s L Ž f . , for every f g Ł nd . Likewise, we can use the fact that ŽUk < Vk . are invertible d to show that LnŽ f . s L Ž f . for all f g Ł nqmq1 . The basis in Ž2.3. contains d ⺡ nq mq1, 2 mq2 , hence, each f g Ł k , k ) n q m q 1, can be written as

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f s g q h, where g g I is a linear combination of the components of x ␣ ⺡ nq mq1, 2 mq2 , < ␣ < s k y n y m y 1, and h is a polynomial of degree k y 1. Hence, we can use induction on k to show that LnŽ f . s L Ž f . as long as x ␣ , < ␣ < s k y n y m y 1, is orthogonal to ⺡ nq mq1, 2 mq2 . By definition, ⺡ nq mq1, 2 mq2 is orthogonal to polynomials of degree at most n q m q 1 y Ž2 m q 2. y 1 s n y m y 2, we conclude that k y n y m y 1 F n y m y 2, or k F 2 n y 1, which completes the proof. Let us comment on the condition of Theorem 2.3. First of all, the d  4 assumption I l Ł ny 1 s 0 is necessary for I to be a generating ideal of a cubature formula of degree 2 n y 1. In fact, if there is a non-zero polynod 2 mial P g Ł ny is 2 n y 2, the cubature 1 , then since the degree of P 2. 2. Ž Ž formula will imply that L P s Ln P s 0, which contradicts the fact d d Ž . that L is square positive. The assumption Ł drI s Ł nq m r I l Ł nqm shows that codim I is finite, and we emphasize the part of m F n y 1. Both these assumptions can be easily verified; in fact, they often follow from the condition < V < s codim I. As Theorem 2.3X shows, they do not impose serious restrictions on I. The essential condition is clearly < V < s codim I, or Ž2.5.. Let us also point out that this condition means that the ideal I satisfies I s I Ž V .. That is, I is to be the same as the largest ideal that contains all polynomials vanishing on V Ž I .. In general, we only have < V < F codim I; the equality is rare. If we are dealing with polynomials in ⺓w x 1 , . . . , x d x, recall that ⺓ is algebraically closed, then we know that < V < s codim I if I is a radical ideal. The fact that we are dealing with Ł d s ⺢w x 1 , . . . , x d x makes the problem more complicated. When does < V < s codim I hold is an essential question about cubature formula of degree 2 n y 1. We will give some necessary conditions later in the section. Let us consider a few special cases in the following, and show in the process how general the above theorem is in comparison to the results in the literature. Case 1. We start with the extreme case that m s n and I s ²⺠n :. It d follows easily that codim I s dim Ł ny 1 . Hence, the theorem states that if d < V < s dim Ł ny 1 , then there is a cubature formula of degree 2 n y 1. This is a result due to Mysovskikh Žcf. w6x., who proved that a cubature formula d of degree 2 n y 1 exists if, and only if, ⺠n has dim Ł ny1 common zeros. Moreover, it is proved in w11x that ⺠n has this many common zeros Žin other words, < V < s codim I . if, and only if, the matrices in the three-term relation Ž2.1. satisfies the condition A ny 1 , i ATny1, j s A ny1, j ATny1, i

1 F i , j F d.

Since matrix multiplication is not commutative, this shows that Gaussian cubature formulae are rare. See w12, 13x for further discussion and examples.

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Case 2. Next we consider the case I s ²U0T ⺠n , ⺡ nq1, 2 :, where U0 is the matrix with rank rnd y ␴ 0 as before and we recall that ⺡ nq 1, 2 s ⺠nq1 q ⌫1⺠n q ⌫2 ⺠ny1 . d It follows that codim I s dim Ł ny 1 q ␴ 0 . Hence, the theorem states that d < < if V s dim Ł ny 1 q ␴ 0 , then there is a cubature formula of degree w3, 4x, who discussed 2 n y 1. The first result of this type is due to Moller ¨ the case of cubature formulae attaining his lower bound for d s 2. For many linear functional L , including standard integrals defined on square, disk Žcentrally symmetric ones., and triangle on ⺢ 2 , Moller proved that ¨ ␴ 0 G w nr2x, where w x x means the largest integer less than or equal to x. That is, < V < G dim Ł 2ny 1 q w nr2x, called Moller’s lower bound. He further ¨ proved that if a cubature formula attains this lower bound, then span x 1U0T ⺠n , x 2 U0T ⺠n4 contains ⺡ nq1, 2 and I s ²U0T ⺠n :, in our language. We note that the condition ⺡ nq 1, 2 ; ²U0T ⺠n : implies by Ž2.2. that ␴ 0 F nr2, so that we have in this case w nr2x F ␴ 0 F nr2. The general case I s ²U0T ⺠n , ⺡ nq1, 2 : is studied in w12x; a complete characterization which answers the question of when < V < s codim I is as follows.

THEOREM 2.4. The ideal I s ²U0T ⺠n , ⺡ nq1, 2 : satisfies < V < s codim I s d dim Ł ny 1 q ␴ 0 if there exists a matrix V such that ⌫1 and V satisfy the following conditions: A ny 1, i Ž VV T y I . ATny1, j s A ny1, j Ž VV T y I . ATny1, i ,

1 F i , j F d,

Ž 2.6a. Ž Bn , i y A n , i ⌫1 . VV T s VV T Ž Bn , i y ⌫1TATn , i . ,

1 F i F d, Ž 2.6b .

VV TATny 1, i A ny1, j VV T q Ž Bn , i y A n , i ⌫1 . VV T Ž Bn , j y ⌫1TATn , j . s VV TATny 1, j A ny1, i VV T q Ž Bn , j y A n , j ⌫1 . VV T Ž Bn , i y ⌫1TATn , i .

Ž 2.6c. for 1 F i, j F d; moreo¨ er, the matrix U is determined by U T V s 0 and the matrix ⌫2 is determined by ⌫2 s Ý dis1 Dn,T i Ž I y VV T . ATny1, i , where Dn, i are matrices that satisfy Ý dis1 Dn,T i A n, i s I. It is also proved in w12x that the cubature formula of degree 2 n y 1 generated by the ideal I s ²U0T ⺠n , ⺡ nq1, 2 : has positive weights. Moreover, if I is a generating ideal of a cubature formula of degree 2 n y 1 with positive weights, then the conditions in Ž2.6. are necessary as well w13x. This shows that we can determine the ideal by solving the equations in Ž2.6.. However, these equations are nonlinear; they are difficult to solve, even

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the existence of a solution is difficult to determine. In the literature, most examples are given for the case when Moller’s lower bound is attained, in ¨ which the ideal I is essentially generated by U0T ⺠n . In w12x, however, one example is found which has ␴ 0 s n for all n G 1, that is, U0T ⺠n contains only one polynomial. We refer to w3᎐5, 7, 8, 12, 13x for some explicit examples and for further discussions. general case I s ² U 0T ⺠ n , U 1T ⺡ n q 1 , 2 , . . . , was briefly considered in w12x, but the emphasis was put on conditions like Ž2.5. ᎐ Ž2.7. which warrants that I has maximal number of common zeros. The conditions are rather complicated and not practical. The existence of the cubature formula is established only when those conditions are satisfied. In contrast to the discussion in w12x, we have shown, in Theorem 2.3, that a cubature formula of degree 2 n y 1 exists as long as I generated by an orthogonal basis satisfies < V < s codim I. The existence of the cubature formula no longer depends on solving the nonlinear system of equations such as those in Ž2.6.. Furthermore, it steps beyond the restriction that I contains ⺡ nq 1, 2 , which implies that Ł drI s Ł ndrI. In practice, if we have a set of orthogonal and quasi-orthogonal polynomials that have a large number of common zeros, then we can check whether they generate a cubature formula of degree 2 n y 1 by examining the condition < V < s codim I. The codimension of I is usually easy to check. The simplest case d is I s ²U0T ⺠n :, for which the codimension of I is simply dim Ł ny1 q ␴0 d q ⭈⭈⭈ q␴m , where ␴ 0 is the rank U0 and ␴ k s rnqk y dim span x ␣ U0T ⺠n : < ␣ < s k 4 . For example, we shall show, in the following section, that the usual product type formula of degree 2 n y 1 in d-variables corresponds to the extreme case that ²U0T ⺠n : s ² P1n, . . . , Pdn :, that is, I is generated by only d orthogonal polynomials of degree n. We finish this section by giving a necessary condition for I to satisfy < V < s codim I. C ase

3.

The

UmT ⺡ nqm, 2 m , ⺡ nqmq1, 2Ž mq1. :

THEOREM 2.5. Let I be the ideal as in Theorem 2.3. If the cubature formula of degree 2 n y 1 generated by I has positi¨ e weights, then there is a nonnegati¨ e definite matrix W such that A ny 1, i Ž W y E . ATny1, j s A ny1, j Ž W y E . ATny1, i ,

Ž 2.7.

where E is the identity matrix, and the matrix U0 satisfies WU0 s 0. Proof. Let Ln denote the positive cubature formula of degree 2 n y 1 generated by I. Assume that U0 has rank rnd y ␴ 0 . It follows that the matrix W [ LnŽ⺠n ⺠nT . is of rank ␴ 0 . Indeed, if the rank of W is less than ␴ 0 , there will be at least rnd y ␴ 0 q 1 linearly independent polynomials

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YUAN XU

aTi ⺠n in I, where a i are solutions of Wa s 0, since LnŽwaTi ⺠n x 2 . s aTi Wa i s 0. This contradicts the fact that U0 has rank ␴ 0 . Since aT Wa s LnŽwaT ⺠n x 2 . G 0 for any a, W is a nonnegative definite matrix. We have U0T W s U0T LnŽ Pn ⺠nT . s 0 since U0T ⺠n vanishes on the variety of I. Since the cubature formula is of degree 2 n y 1, we have LnŽ⺠n ⺠kT . s T . L Ž⺠n ⺠kT . s 0 for k F n y 1 by orthogonality, and LnŽ x i ⺠ny1⺠ny1 s T L Ž x i ⺠ny1⺠ny1 . s Bny1, i , where the second equality follows from applying L on both sides of the three-term relation. Similarly, we have T . InŽ x i ⺠ny2 ⺠ny1 s A ny2, i Žsee, for example, w13x.. Together with LnŽ⺠n ⺠nT . s W, this allows us to use the three-term relation Ž2.1. to compute the T . matrix LnŽ x i x j ⺠ny1⺠ny1 . We have T Ln Ž x i x j ⺠ny1⺠ny1 .

s Ln Ž A ny1, i ⺠n q Bny1, i ⺠ny1 q A ny2, i ⺠ny2 . = Ž A ny 1, j ⺠n q Bny1, j ⺠ny1 q A ny2, j ⺠ny2 .

T

s A ny 1, iWA ny1, j q Bny1, i Bny1, j q ATny2, i A ny2, j Since the left hand side remains the same if we switch the order of i and j, the above equation implies Ž2.7. upon applying one of the commuting conditions satisfied by the coefficient matrices of the three-term relation Ž2.1. Žcf. w11x or w13x.. This theorem shows that if the cubature formula is positive, then the generating ideal I satisfies the necessary condition Ž2.7.. It is easy to see that Ž2.7. and Ž2.6a. are in fact the same. A proof that Ž2.6a. is necessary in Theorem 2.4 is outlined in w13x, which we have followed in the above proof. For two-dimensional cubature formulae, that is, d s 2, the condition Ž2.7. can be made more explicit. We need the following notation. For all monomials of order n, we define a column vector x n s Žx ␣ . < ␣
y1

LTny1, j s L ny1, j Gny1 Ž W y E . Ž GnT .

y1

LTny1, i .

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225

For d s 2, the matrices L n, i are simply L n, 1 s Ž E <0. and L n, 2 s Ž0 < E .. Hence, it is easy to see that the above equation is equivalent to the fact that Gny1 ŽW y E .Ž GnT .y1 is a Hankel matrix. We state the result as a corollary. COROLLARY 2.6. Let d s 2 and let I be as in Theorem 2.6. Then there is a Hankel matrix H s Ž h iqj . such that the matrix E q Gn HGnT is nonnegati¨ e definite and U0 satisfies Ž E q Gn HGnT .U0 s 0. In particular, the matrix E q Gn HGnT has rank ␴ 0 . We remark that the matrix Gn can often be written down explicitly. For example, let w 1 and w 2 be nonnegative weight functions defined on the interval w a, b x and w c, d x, respectively, and let pnŽ x . s ␥n x n q ⭈⭈⭈ and qnŽ x . s ␶n x n q ⭈⭈⭈ denote the n-th order orthonormal polynomials associated to w 1 and w 2 , respectively. Then a basis of orthonormal polynomials with respect to the product weight function W Ž x, y . s w 1Ž x . w 2 Ž y . on w a, b x = w c, d x is given by pny k Ž x . pk Ž y .. If we order the monomials of degree n by x n, x ny 1 y, . . . , y n, then the matrix Gn is a diagonal matrix defined by Gn s diag␥n␶ 0 , ␥ny1␶ 1 , . . . , ␥ 1␶ny1 , ␥ 0␶n4 . Similar result holds for orthogonal polynomials in a disk or a triangle Žsee w12x.. The coefficient matrices A n, i in the three-term relation Ž2.1. can be computed explicitly y1 using A n, i s Gn L n, i Gnq1 . We can use A n, i to help compute the codimension of I, or the right hand of Ž2.5., by the use of the basic relation Ž2.2.. For example, if I s ²U T ⺠n :, then rank U s dim U0 and rank Uj T s dim Uj for j G 1, where U1 s

U TA n , 1 U TA n , 2

,

U2 s

U1TA nq1, 1 U1TA nq1, 2

,... .

Recall that the Gaussian cubature formula, if it exists, is generated by ⺠n , that is, by all orthogonal polynomials of degree n. In general, to get a cubature formula that resembles Gaussian cubature, we may expect that the ideal I is generated by as many orthogonal polynomials as possible, which means that E q Gn HGnT in Corollary 2.6 has as small a rank as possible. On the other hand, the rank cannot be too small, since Moller’s ¨ lower bound gives a lower bound for ␴ 0 . Thus, for example, for the centrally symmetric linear functional, we have ␴ 0 G w nr2x. To choose a Hankl matrix such that E q Gn HGnT has a relative small rank and is nonnegative definite is not always easy. The existence of such a Hankl matrix is only a necessary condition for I to generate a cubature formula. However, our result states that if we can find a matrix U such that U T ⺠n has a large number of common zeros, then we can compute codim I and establish the existence of a cubature formula by Theorem 2.3. It is in this

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YUAN XU

regard that the necessary condition in Corollary 2.6 is useful. We shall illustrate some aspects of these results by examples in the following section.

3. EXAMPLES As we pointed out in the previous section, many cubature formulae in the literature, in particular, those discussed in Case 1 and Case 2, can be taken as applications of Theorem 2.3. In this section, we give several other examples that illustrate Theorem 2.3. To help check the conditions, we introduce the following notation. With respect to our fixed monomial order, we denote by LM Ž f . the leading monomial for any polynomial f g Ł d. That is, if f s Ýc␣ x ␣ , then LM Ž f . s x ␤, where x ␤ is the leading monomial among all monomials in f with respect to the monomial order. For an ideal I in Ł d other than  04 , we denote by LM Ž I . the leading terms of I, that is, LM Ž I . s  x ␣
Hwy1, 1x

d

f Ž x . W Ž x . dx s

Ý

k 1s1

n

⭈⭈⭈

Ý

k ds1

␭ k 1 , 1 ⭈⭈⭈ ␭ k d , d f Ž x i , 1 , . . . , x i , d . ,

where for each i, the number ␭ k, i and x i, k are the weights and nodes of

227

CUBATURE FORMULAE AND POLYNOMIAL IDEALS

Gaussian quadrature formula of degree 2 n y 1 with respect to wi . Let pnŽ wi . denote the nth orthonormal polynomial with respect to wi . Then the nodes x 1, i , . . . , x n, i are the zeros of pnŽ wi .. A basis of orthonormal polynomials for W on wy1, 1x d is given by Pk Ž x . s pk 1Ž x 1 . ⭈⭈⭈ pk dŽ x d . ,

k s Ž k1 , . . . , k d . ,

where the degree of Pk is
kr2

CkŽ ␮ . y Ž 1 y x 2 .

ž

y1r2

0 F k F n,

/, Ž 3.1.

where CkŽ ␭. denotes the Gegenbauer polynomial of degree k, orthogonal with respect to Ž1 y t 2 . ␭y1r2 on wy1, 1x, and A nk are normalization constants Žcf. w2x.. We consider I s P0n , Pnn s CnŽ ␮ q1r2. Ž x . , Ž 1 y x 2 .

¦

; ¦

nr2

CnŽ ␮ . y Ž 1 y x 2 .

ž

y1r2

/;.

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YUAN XU

The variety of I is V s Ž x i , yi Ž1 y x i2 .1r2 . N 1 F i, j F n4 , where x 1 , . . . , x n are zeros of CnŽ ␮ q1r2. and y 1 , . . . , yn are zeros of CnŽ ␮ ., hence, < V < s n2 . We compute the codimension of I. If we choose a graded monomial order such that the monomials of degree n are arranged in the order of y n, xy ny1, . . . , x ny1 y, x n, then it becomes clear that LM Ž P0n . s x n and LM Ž Pnn . s y n. Hence, we see that LM Ž I . contains  x k y m N k q m s 2 n y 14 just as in Example 1. We conclude that Ł drI s Ł 2d ny1rI and codim I s n2 s < V <. The cubature formula of degree 2 n y 1 based on the points in V is the usual product formula, n

HB

2

f Ž x, y . W␮ Ž x, y . dx dy s

n

Ý Ý ␭i ␮ j f ž x i , yi Ž 1 y x i2 .

1r2

is1 js1

/,

f g Ł 22 ny1 , where ␭ i and ␮ i are weights of Gaussian quadrature formula of degree 2 n y 1 with respect to Ž1 y t 2 . ␮ and Ž1 y t 2 . ␮y1r2 , respectively Žsee, for example, w6, 8x.. The case of product formula on the simplex works similarly. We consider the weight function W Ž x, y . s x ␣ y ␤ Ž1 y x y y .␥ on the triangle that has vertices at Ž0, 0., Ž1, 0. and Ž0, 1.. A basis of orthogonal polynomials with respect to W is given by Ž2 kq ␤ q ␥ , ␣ y1r2. Pkn Ž x, y . s Bkn Pnyk Ž 2 x y 1.

= Ž 1 y x . PkŽ ␥y1r2, ␤y1r2. 2 k

ž

y 1yx

y1 ,

/

for 0 F k F n, where PkŽ ␭ , ␮ . denotes the Jacobi polynomial of degree k, orthogonal with respect to Ž1 y t . ␭Ž1 q t . ␮ on wy1, 1x, and Bkn are normalization constants Žcf. w2x.. Considering I s ² P0n, Pnn :, we can show < V < s codim I just as in the case of disk. The cubature formula of degree 2 n y 1 is the usual product formula on the triangle. EXAMPLE 3 ŽOther Cubature Formulae on Disk and on Triangle with n2 Nodes.. In the previous examples, we showed that the usual product type formulae in two dimension can be constructed using an extreme case of I s ² P1n, Pnn : of the Theorem 2.3. We can choose two different polynomials in I and generate a cubature formula of degree 2 n y 1 with n2 nodes which is nevertheless different from the product formula. We work with the weight function W␮ on the unit disk B 2 first. The nth reproducing kernel function PnŽx, y. for a system of orthogonal polynomials is defined

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229

by Pn Ž x, y . s

Ý

< ␣
P␣n Ž x . P␣n Ž y . .

We recall a compact formula for the reproducing kernel for W␮ , ␮ G 0, on B d, Pn Ž x, y . s

n q ␮ q Ž d y 1 . r2

␮ q Ž d y 1 . r2 = x⭈yq

ž

1

Hy1 C

Ž ␮ qŽ dy1.r2. n

'1 y
=Ž 1 y t 2 .

2

␮y1

dt

2

1

Hy1 Ž 1 y t

2

.

␮ y1

dt,

x, y g B d ,

for ␮ s 0 the above formula holds upon taking limit ␮ ª 0 Žsee w12x.. From the definition of PnŽx, y., for each fixed a, the polynomial PnŽx, a. is an orthogonal polynomial of degree n; in particular, the polynomial CnŽ ␮ qŽ dy1.r2. Ž x i . s const PnŽx, e i . is an orthogonal polynomial for each i, where e 1 , . . . , e d forms the usual Euclidean basis as before. We consider the ideal I s² CnŽ ␮ qŽ dy1.r2. Ž x 1 . , . . . , CnŽ ␮ qŽ dy1.r2. Ž x d .: . Evidently, the variety of I is given by V s Ž x k, 1 , . . . , x k, d . N 1 F k F n, 1 F i F d4 , where x k, i are zeros of CnŽ ␮ qŽ dy1.r2. on Žy1, 1.. Hence, < V < s n d. On the other hand, since LM Ž I . contains x in for 1 F i F d, it follows that codim I s n d as in Example 1. Therefore, by Theorem 2.3 there is a cubature formula of degree 2 n y 1 that takes the form n

HB

d

f Ž x . W␮ Ž x . dx s

Ý

k 1s1

n

⭈⭈⭈

Ý

k ds1

␭k1 , . . . , k d f Ž x i , 1 , . . . , x i , d . .

The curious thing about this formula is that its nodes are the same as the d Ž product formula for Ł is1 1 y x i . ␮qŽ dy2.r2 on wy1, 1x d, yet it is a cubature d formula for W␮ on B . We note that some of the nodes are outside of the region B d. By choosing different a i in the reproducing kernel, we can consider other ideals generated by d orthogonal polynomials which lead to still different cubature formulae of degree 2 n y 1 based on n d common zeros. There is a similar compact formula for the reproducing kernel with respect

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YUAN XU

to the Jacobi type weight function on the simplex Ý d in ⺢ d, which can be used to generate cubature formulae of degree 2 n y 1 on Ý d that uses n d nodes but different from the product type formula. Since the procedure is similar, we shall not elaborate. In the previous examples, our ideals are generated by only d polynomials, which corresponds to the extreme case of Theorem 2.3. For d s 2, the ideal is generated by 2 polynomials, which is as few generators as it can possibly be while still having zero dimensional variety. We also note that by Bezout’s theorem, the two polynomials can have at most n2 common zeros. In the following examples, we consider the case where I has more generators. EXAMPLE 4 ŽCubature Formula of Degree 9 on a Disk.. In this example, we consider cubature formulae with respect to the Lebesgue measure Žunit weight function. on the unit disk. That is, the integral is defined by L Ž f . s HB 2 f Žx. dxr␲ . We take a basis of orthonormal polynomials Pkn as in Ž3.1. with ␮ s 0. The first case that requires serious computation is n s 5, or cubature formulae of degree 9. Most of the computation below is carried out using the computer algebra system Mathematica. We start with constructing a Hankl matrix that satisfies the condition in Corollary 2.7. Instead of E q Gn HGnT, we can look at the matrix Gny1 Ž GnT .y1 q H. The matrix G5 can be computed from the leading coefficients of the polynomials in Ž3.1., 32 0 G5 s

0 32 15r7

0 0

0 0

0 0

0 0

40 2r7

0

120 2r7

'

0

0

0

0

8'30

0

40 10r3

0

0

6 6r7

0

60 6r7

0

10'42

0

0

10 6r7

0

20 14r3

0

6'42

'

'

'

'

'

'

'

.

We choose H s Ž h iqj . 5i, js0 to have h1 s h 3 s h 5 s h 7 s h 9 s 0. It turns Ž G5T .y1 q H has the same form as H, that is, its out that the matrix Gy1 5 zero elements are in the same positions as the zero elements of H. The form shows that the rows 1, 3, 5 and rows 2, 4, 6 are linearly independent. We choose to have the submatrix of the three odd rows rank 2 and the submatrix of the three even rows rank 1. Setting row2 s s ⭈ row4 q t ⭈ row6,

CUBATURE FORMULAE AND POLYNOMIAL IDEALS

231

the matrix H is determined in terms of s and t. Solving the equation Ž E q G5 HG5T .U s 0 for U, we find three polynomials in U T ⺠n as follows, Q15 Ž x, y . s y

63t q 5s Ž 14 q 3t . 24'10 st

(

Q25 Ž x, y . s y

q

P15 Ž x, y . q P55 Ž x, y . ,

7 Ž 35 y 19t . t q s Ž 94 q 69t q 13t 2 . 8 Ž 8t 2 q s Ž y3 q 7t q 4 t 2 . .

6

t Ž y35 q 43t . q s Ž y94 y 21t q 11t 2 . 4'3 Ž 8t 2 q s Ž y3 q 7t q 4 t 2 . .

P05 Ž x, y . P25 Ž x, y .

q P45 Ž x, y . ,

(

Q35 Ž x, y . s y

7 Ž 5 q 3t . 6t

2

P15 Ž x, y . q P35 Ž x, y . .

At this point we need to choose s and t so that Q15, Q35 and Q55 have a large amount of common zeros. There is usually no easy way to determine which s and t to choose. An arbitrary choice will not work; for example, the case s s 1 and t s 2 leads to only 5 common zeros. It turns out, however, that Q25 becomes independent of s and t when t q sŽ t q 2. s 0. So, we take s s ytrŽ t q 2., which leads to Q25 Ž x, y . s

(

7 6

P05 Ž x, y . q

2

'3

P25 Ž x, y . q P45 Ž x, y . .

With the help of Mathematica, we find that Q15 and Q25 have 25 distinct common zeros for every t. The common zeros are given by algebraic expressions of t, some involving two folds of square roots; we shall not give them here. For t between 27r34 F t F 9 q 4'6 , all 25 common roots are real. Upon computing codimension of ² Q15, Q25 :, we conclude that there is a cubature formula of degree 9 with 25 points for each t in the range w27r34, 9 q 4'6 x. More interestingly, we find that Q35 vanishes on 21 of the 25 points for t in the above range, except when t s 9 q 4'6 in which case Q35 vanishes on 23 points. Checking the codimension of I s ² Q15, Q25 , Q35 :, it turns out that we have ␴ 0 s 3, ␴ 1 s 2, ␴ 2 s 1, and ␴ 3 s 0 for all t. Hence, codim I s 15 q 3 q 2 q 1 s 21. Thus, by Theorem 2.3, we conclude that there is a cubature formula of degree 9 with 21 nodes for each t in the range w27r34, 9 q 4'6 .. In the above example, by adding Q35 to the ideal generated by Q15 and 5 Q2 to form a new ideal, we reduce the size of the variety and the

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YUAN XU

codimension at the same time. One may try to do the same for n ) 5 starting with one of the examples with < V < s n2 . If we know the common zeros, we can compute any polynomial that vanishes on them. In general, however, we do not know which zeros can be reduced, which prevents us to compute additional polynomial that vanishes on part of the common zeros. This shows that the necessary condition in Corollary 2.6 is useful. There is yet another situation for which the procedure described above is indeed useful; it is explained in our next example. EXAMPLE 5 ŽAnother Cubature Formula of Degree 9 on the Unit Disk.. We consider another cubature formula of degree 9 for the Lebesgue measure on the unit disk. This formula was discovered by Mysovskikh Žsee w6, formula 29 for the ballx., which uses only 19 nodes and it was shown that the nodes are common zeros of two orthogonal polynomials, given in terms of our basis by Q15 Ž x, y . s 3 P15 Ž x, y . q 2'14 P35 Ž x, y . q 5'10 P55 Ž x, y . , Q25 Ž x, y . s

4 q '6 60 q

P05 Ž x, y . y

y1 q '6 10'42

11'14 q 13'21 840

P25 Ž x, y .

P45 Ž x, y . .

It is not difficult to find all common zeros Žthey are given in w6x., which allows us to find out that there is one more orthogonal polynomial of degree 5 vanishing on all 19 nodes, which is given by Q35 Ž x, y . s

(

7 6

P05 Ž x, y . q

2

'3

P25 Ž x, y . q P45 Ž x, y . .

However, it is easily shown that the codimension of ² Q15, Q25 , Q35 : is again 21. According to Theorem 2.3, this shows that the ideal should be enlarged by adding more elements to its basis; moreover, the polynomial to be added should be a quasi-orthogonal polynomial of degree 6 and order 2. Since we know all the common zeros, we are able to find such a polynomial. It turns out to be an orthogonal polynomial of degree 6. Given in terms of our basis Ž3.1., this polynomial, denoted by Q16 , is given by Q16 Ž x, y . s

3 Ž y491 q 201'6 . 4 Ž 157'2 y 129'3 . q P56 Ž x, y . .

P16 Ž x, y . q

'7 Ž 173 y 72'6 . 6 P3 Ž x, y . 20 Ž 4'2 y 3'3 .

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Moreover, it is easy to verify that Q16 is not in the ideal ² Q15, Q25 , Q35 :. Adding it to the ideal and checking the codimension, we finally conclude that the cubature formula is generated by the ideal ² Q15, Q25 , Q35, Q16 :. There is a notable difference between this example and the previous one. In this example, the polynomials Q15 and Q25 have 19 common zeros instead of 25. Since we no longer need to reduce zeros in the variety, we found additional polynomials that vanish on all common zeros without difficulty. We would like to use such a procedure in other possible situation. In fact, if we can find two or more orthogonal polynomials whose number of common zeros is less than n2 and greater than, say, dim Ł 2ny1 q w nr2x, then we can compute additional orthogonal and quasi-orthogonal polynomials so that we get an ideal large enough to satisfy codim I s < V <, thus, showing that a cubature formula exists. The essential question is how to find two or more orthogonal polynomials so that they have a large number of common zeros. The necessary condition in Theorem 2.5 or Corollary is helpful in this regard.

REFERENCES 1. D. Cox, J. Little, and D. O’Shea, ‘‘Ideals, Varieties, and Algorithms,’’ 2nd ed., SpringerVerlag, Berlin, 1997. 2. T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in ‘‘Theory and Application of Special Functions’’ ŽR. A. Askey, Ed.., pp. 435᎐495, Academic Press, New York, 1975. 3. H. M. Moller, ‘‘Polynomideale und Kubaturformeln,’’ Thesis, Univ. Dortmund, 1973. ¨ 4. H. M. Moller, Kubaturformeln mit minimaler Knotenzahl, Numer. Math. 35 Ž1976., ¨ 185᎐200. 5. C. R. Morrow and T. N. L. Patterson, Construction of algebraic cubature rules using polynomial ideal theory, SIAM J. Numer. Anal. 15 Ž1978., 953᎐976. 6. I. P. Mysovskikh, Numerical characteristics of orthogonal polynomials in two variables, Vestn. Leningrad Uni¨ . Math. 3 Ž1976., 323᎐332. 7. I. P. Mysovskikh, The approximation of multiple integrals by using interpolatory cubature formulae, in ‘‘Quantitative Approximation’’ ŽR. A. DeVore and K. Scherer, Eds.., Academic Press, New York, 1980. 8. I. P. Mysovskikh, ‘‘Interpolatory Cubature Formulas,’’ Nauka, Moscow, 1981 win Russianx. 9. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 Ž1948., 286᎐300. 10. A. Stroud, ‘‘Approximate Calculation of Multiple Integrals 7,’’ Prentice-Hall, Englewood Cliffs, NJ, 1971. 11. Y. Xu, Block Jacobi matrices and zeros of multivariate orthogonal polynomials, Trans. Amer. Math. Soc. 342 Ž1994., 855᎐866. 12. Y. Xu, ‘‘Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature,’’ Pitman Research Notes in Mathematics Series, Longman, Essex, 1994. 13. Y. Xu, On orthogonal polynomials in several variables, in ‘‘Special Functions, q-Series, and Related Topics,’’ Fields Institute Communications, Vol. 14, pp. 247᎐270, 1997. 14. Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in ⺢ d, Trans. Amer. Math. Soc. 351 Ž1999., 2439᎐2458.