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Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls∗

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Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls∗ SIAM J. MATH. ANAL. c 1998 Society for Industrial and Applied Mathematics Vol. 29, No. 3, pp. 779{793, May 1998 015 ORTHOGONAL POLYNOMIALS AND CUBATURE FORMULAE ON SPHERES AND ON BALLS∗ YUAN XUy Abstract. Orthogonal polynomials on the unit sphere in Rd+1 and on the unit ball in Rd are shown to be closely related to each other for symmetric weight functions. Furthermore, it is shown that a large class of cubature formulae on the unit sphere can be derived from those on the unit ball and vice versa. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres. Key words. orthogonal polynomials in several variables, on spheres, on balls, spherical har- monics, cubature formulae AMS subject classifications. 33C50, 33C55, 65D32 PII. S0036141096307357 1. Introduction. We are interested in orthogonal polynomials in several vari- ables with emphasis on those orthogonal with respect to a given measure on the unit sphere Sd in Rd+1. In contrast to orthogonal polynomials with respect to measures defined on the unit ball Bd in Rd, there have been relatively few studies on the struc- ture of orthogonal polynomials on Sd beyond the ordinary spherical harmonics which are orthogonal with respect to the surface (Lebesgue) measure (cf. [2, 3, 4, 5, 6, 8]). The classical theory of spherical harmonics is primarily based on the fact that the ordinary harmonics satisfy the Laplace equation. Recently Dunkl (cf. [2, 3, 4, 5] and the references therein) opened a way to study orthogonal polynomials on the spheres with respect to measures invariant under a finite reflection group by developing a theory of spherical harmonics analogous to the classical one. In this important the- ory the role of Laplacian operator is replaced by a differential-difference operator in the commutative algebra generated by a family of commuting first-order differential- difference operators (Dunkl's operators). Other than these results, however, we are not aware of any other method of studying orthogonal polynomials on spheres. A closely related question is constructing cubature formulae on spheres and on balls. Cubature formulae with a minimal number of nodes are known to be related to orthogonal polynomials. Over the years, a lot of effort has been put into the study of cubature formulae for measures supported on the unit ball, or on other geometric domains with nonempty interior in Rd. In contrast, the study of cubature formulae on the unit sphere has been more or less focused on the surface measure on the sphere; there is little work on the construction of cubature formulae with respect to other measures. This is partly due to the importance of cubature formulae with respect to the surface measure, which play a role in several fields in mathematics, and perhaps partly due to the lack of study of orthogonal polynomials with respect to a general measure on the sphere. One main purpose of this paper is to provide an elementary approach towards the study of orthogonal polynomials on Sd for a large class of measures. This approach is ∗ Received by the editors July 26, 1996; accepted for publication (in revised form) January 9, 1997. This research was supported by the National Science Foundation under grant DMS-9500532. http://www.siam.org/journals/sima/29-3/30735.html yDepartment of Mathematics, University of Oregon, Eugene, OR 97403-1222 (yuan@math. uoregon.edu). 779 780 YUAN XU based on a close connection between orthogonal polynomials on Sd and those on the unit ball Bd; a prototype of the connection is the following elementary example. For d = 1, the spherical harmonics of degree n are given in the standard polar coordinates by (1) n (2) n (1.1) Yn (x1;x2)=r cos nθ and Yn (x1;x2)=r sin nθ: Under the transform x = cos θ, the polynomials Tn(x) = cos nθ and Un(x) = sin nθ= sin θ are the Chebyshev polynomials of the first and the second kind, orthogonal with re- spect to 1=p1 x2 and p1 x2, respectively, on the unit ball [ 1; 1] in R. Hence, the spherical harmonics− on S−1 can be derived from orthogonal polynomials− on B1. We shall show that for a large class of weight functions on Rd+1 we can con- struct homogeneous orthogonal polynomials on Sd from the corresponding orthog- onal polynomials on Bd in a similar way. This allows us to derive properties of orthogonal polynomials on Sd from those on Bd; the latter have been studied much more extensively. Although the approach is elementary and there is no differential or differential-difference operator involved, the result offers a new way to study the structure of orthogonal polynomials on Sd. Our approach depends on an elementary formula that links the integration on Bd to the integration on Sd. The same formula yields an important connection between cubature formulae on Sd and those on Bd; the result states roughly that a large class of cubature formulae on Sd is generated by cubature formulae on Bd and vice versa. In particular, it allows us to shift our attention from the study of cubature formulae on the unit sphere to the study of cubature formulae on the unit ball; there has been much more understanding towards the structure of the latter one. Although the result is simple and elementary, its importance is apparent. It yields, in particular, many new cubature formulae on spheres and on balls. Because the main focus of this paper is on the relation between orthogonal polynomials and cubature formulae on spheres and those on balls, we will present examples of cubature formulae in a separate paper. The paper is organized as follows. In section 2 we introduce notation and present the necessary preliminaries, where we also prove the basic lemma. In section 3 we show how to construct orthogonal polynomials on Sd from those on Bd. In section 4 we discuss the relation between cubature formulae on the unit sphere and those on the unit ball. 2. Preliminary and basic lemma. For x; y Rd we let x y denote the usual 2 · inner product of Rd and x =(x x)1=2 the Euclidean norm of x. Let Bd be the unit j j · ball of Rd and Sd be the unit sphere on Rd+1; that is, d d d d+1 B = x R : x 1 and S = y R : y =1 : f 2 j |≤ g f 2 j j g Polynomial spaces. Let N0 be the set of nonnegative integers. For α =(α1,...,αd) d d α α1 αd 2 N0 and x =(x1;:::;xd) R we write x = x1 xd . The number α 1 = 2 α ··· d j j α1 + + αd is called the total degree of x . We denote by Π the set of polynomials ··· d d in d variables on R and by Πn the subset of polynomials of total degree at most n. d d We also denote by n the space of homogeneous polynomials of degree n on R and we let rd = dim d.P It is well known that n Pn n + d n + d 1 dim Πd = and rd = : n n n n − Orthogonal polynomials on Bd. Let W be a nonnegative weight function on Bd and assume d W (x)dx < . It is known that for each n N0 the set of polynomials B 1 2 R ORTHOGONAL POLYNOMIALS AND CUBATURE FORMULAE 781 of degree n that are orthogonal to all polynomials of lower degree forms a vector space d n d n whose dimension is rn. We denote by Pk ,1 k rn and n N0, one family of V f g ≤d ≤ 2 d orthonormal polynomials with respect to W on B that forms a basis of Πn, where the superscript n means that P n Πd . The orthonormality means that k 2 n n m Pk (x)Pj (x)W (x)dx = δj;kδm;n: d ZB n n T n A useful notation is n =(P1 ;:::;P d ) , which is a vector with P as components P rn j n d (cf. [22, 24]). For each n N0, the polynomials P ,1 k rn, form an orthonormal 2 k ≤ ≤ basis of n. We note that there are many bases of n;ifQ is an invertible matrix of d V V size rn, then the components of QPn form another basis of n which is orthonormal if Q is an orthogonal matrix. For general results on orthogonalV polynomials in several variables, including some of the recent development, we refer to the survey [24] and the references therein. One family of weight functions on Bd whose corresponding 2 µ 1=2 orthogonal polynomials have been studied in detail is (1 x ) − , µ 0, which we will refer to as classical orthogonal polynomials on Bd−|(cf.j [1, 6, 25]). ≥ Ordinary spherical harmonics. The harmonic polynomials on Rd are the homo- geneous polynomials satisfying the Laplace equation ∆P = 0, where 2 2 d ∆=@1 + + @d on R ··· and @i is the ordinary partial derivative with respect to the ith coordinate. They d d d d span a subspace n = ker ∆ n of dimension dim n dim n 2. The spherical H \P P − d 1 P − d harmonics are the restriction of harmonic polynomials on S − .IfYn n, then Yn d 2H d 1 is orthogonal to Q k ,0 k<n, with respect to the surface measure d! on S − . 2P ≤ d Dunkl's h-harmonics. For a nonzero vector v R we define the reflection σv by 2 2 d xσv := x 2(x v)v= v ; x R : − · j j 2 d Suppose that G is a finite reflection group on R with the set vi : i =1; 2; :::; m of f g positive roots; assume that vi = vj whenever σi is conjugate to σj in G, where we j j j j write σi = σv ,1 i m.
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