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Spherical Harmonics Are the Restrictions of Elements in N to the Unit Sphere

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Spherical Harmonics Are the Restrictions of Elements in N to the Unit Sphere SPHERICAL HARMONICS FENG DAI AND YUAN XU Abstract. This is Chapter 1 of the book Approximation Theory and Har- monic Analysis on Spheres and Balls by the authors. It provides a self- contained introduction to spherical harmonics. The book will be published as a title in Springer Monographs in Mathematics by Springer in 2013. The table of contents of the book is attached at the end of this file. In this chapter we introduce spherical harmonics and study their properties. Most of the material of this chapter, except the last section, is classical. We strive for a succinct account of the theory of spherical harmonics. After a standard treat- ment of the space of spherical harmonics and orthogonal bases in the first section, the orthogonal projection operator and reproducing kernels, also known as zonal harmonics, are developed in greater details in the second section, because of their central role in harmonic analysis and approximation theory. As an application of the addition formula, it is shown in the third section that there exist bases of spherical harmonics consisting of entirely zonal harmonics. The Laplace–Beltrami operator is discussed in the fourth section, where an elementary and self-contained approach is adopted. Spherical coordinates and an explicit orthonormal basis of spherical harmonics in these coordinates are presented the fifth section. These formulas in two and three variables are collected in the sixth section for easy reference, since they are most often used in applications. The connection to group representation is treated briefly in the seventh section. The last section deals with derivatives and integrals on the sphere. With the introduction of angular derivatives that are first- order differential operators acting on the large circles of intersections of the sphere and the coordinate planes, it is shown that the Laplace–Beltrami operator can be decomposed into second-order angular derivatives. These derivative operators will play an important role in approximation theory on the sphere. They are used to derive several integral formulas on the sphere. arXiv:1304.2585v1 [math.CA] 9 Apr 2013 1. Space of spherical harmonics and orthogonal bases We begin by introducing some notation that will be used throughout this book. d d For x R , we write x = (x1,...,xd). The inner product of x, y R is denoted ∈ d ∈ N by x, y := i=1 xiyi and the norm of x is denoted by x := x, x . Let 0 h i k k Nd h i α denote the set of nonnegative integers. For α = (α1,...,αd) 0, a monomial x P α1 α p is a product xα = x ...x d , which has degree α = α + ...∈+ α . 1 d | | 1 d Date: April 10, 2013. 2000 Mathematics Subject Classification. 42B10, 42C10. Key words and phrases. spherical harmonics, unit sphere, zonal harmonics, Laplace–Beltrami operator. The work of the first author was supported in part by NSERC Canada under grant RGPIN 311678-2010. The work of the second author was supported in part by NSF Grant DMS-1106113. 1 2 FENG DAI AND YUAN XU A homogeneous polynomial P of degree n is a linear combination of monomials of α degree n, that is, P (x)= |α|=n cαx , where cα are either real or complex numbers. A polynomial of (total) degree at most n is of the form P (x) = c xα. Let P |α|≤n α d denote the space of real homogeneous polynomials of degree n and let Πd denote n P n theP space of real polynomials of degree at most n. Counting the cardinalities of α Nd : α = n and α Nd : α n shows that { ∈ 0 | | } { ∈ 0 | |≤ } n + d 1 n + d dim d = − and dim Πd = . Pn n n n Let ∂i denote the partial derivative in the i-th variable and ∆ the Laplacian operator ∆ := ∂2 + + ∂2. 1 ··· d d Definition 1.1. For n = 0, 1, 2,... let n be the linear space of real harmonic polynomials, homogeneous of degree n, onHRd, that is, d := P d : ∆P =0 . Hn ∈Pn d Spherical harmonics are the restrictions of elements in n to the unit sphere. If d n ′ ′ ′ HSd−1 Y n, then Y (x)= x Y (x ) where x = x x and x . Strictly speaking, ∈ H k k d k k ∈ one should make a distinction between n and its restriction to the sphere. We d H will, however, also call n the space of spherical harmonics. When it is necessary H d to emphasize the restriction to the sphere, we shall use the notation Sd−1 . In Hn| Sd−1 d − Sd−1 d − the same vein, we shall define n( ) := n Sd 1 and Πn( ) := Πn Sd 1 . Spherical harmonics of differentP degrees areP | orthogonal with respect to| 1 (1.1) f,g Sd−1 := f(x)g(x)dσ(x) h i ω Sd−1 d Z d−1 where dσ is the surface area measure and ωd denotes the surface area of S , 2πd/2 (1.2) ωd := dσ = . Sd−1 Γ(d/2) Z d d Theorem 1.2. If Y , Y , and n = m, then Y , Y Sd−1 =0. n ∈ Hn m ∈ Hm 6 h n mi ∂ Proof. Let ∂r denote the normal derivative. Since Yn is homogeneous, Yn(x) = n ′ ′ ′ Sd−1 ∂Yn ′ ′ ′ Sd−1 r Yn(x ), where x = rx and x , so that ∂r (x ) = nYn(x ) for x and n 0. By Green’s identity, ∈ ∈ ≥ ∂Yn ∂Ym (n m) YnYmdσ = Ym Yn dσ − Sd−1 Sd−1 ∂r − ∂r Z Z = Ym∆Yn Yn∆Ym dx =0, Bd − Z since ∆Yn = 0 and ∆Ym = 0. Theorem 1.3. For n =0, 1, 2,..., there is a decomposition of d, Pn (1.3) d = x 2j d . Pn k k Hn−2j 0≤Mj≤n/2 In other words, for each P d, there is a unique decomposition ∈Pn (1.4) P (x)= x 2j P (x) with P d . k k n−2j n−2j ∈ Hn−2j 0≤Xj≤n/2 3 d d d d Proof. The proof uses induction. Evidently 0 = 0 and 1 = 1. Since ∆ d d , dim d dim d dim d .P SupposeH the statementP H holds for Pn ⊂ Pn−2 Hn ≥ Pn − Pn−2 m =0, 1,...,n 1. Then x 2 d is a subspace of d and it is isomorphic to d . − k k Pn−2 Pn Pn−2 By the induction hypothesis, x 2 d = x 2j+2 d . Hence, by k k Pn−2 0≤j≤n/2−1 k k Hn−2−2j the previous theorem, d is orthogonal to x 2 d , so that dim d +dim d Hn Lk k Pn−2 Hn Pn−2 ≤ dim d. Consequently, d = d x 2 d . Pn Pn Hn ⊕k k Pn−2 Corollary 1.4. For n =0, 1, 2,..., n + d 1 n + d 3 (1.5) dim d = dim d dim d = − − , Hn Pn − Pn−2 n − n 2 − where it is agreed that dim d =0 for n =0, 1. Pn−2 Corollary 1.5. For n N, Π (Sd−1)= (Sd−1) (Sd−1) and ∈ n Pn ⊕Pn−1 n + d 1 n + d 2 (1.6) dim Π (Sd−1) = dim d + dim d = − + − . n Pn Pn−1 n n 1 − Sd−1 d Proof. By Theorem 1.3, Πn( ) can be written as a direct sum of k for 0 k n, which gives the stated decomposition by (1.3). Moreover, H ≤ ≤ n n dim Π (Sd−1)= dim d = (dim d dim d ) n Hk Pk − Pk−2 kX=0 Xk=0 by (1.5), which simplifies to (1.6). The orthogonality and homogeneity define spherical harmonics. Proposition 1.6. If P is a homogeneous polynomial of degree n and P is orthog- d onal to all polynomials of degree less than n with respect to , Sd−1 , then P . h· ·i ∈ Hn d Proof. Since P n, P can be expressed as in (1.4). The orthogonality then shows that P = P ∈Pd . n ∈ Hn Let O(d) denote the orthogonal group, the group of d d orthogonal matrices, and let SO(d)= g O(d) : det g =1 be the special orthogonal× group. A rotation in Rd is determined{ ∈ by an element in }SO(d). d Theorem 1.7. The space n is invariant under the action f(x) f(Qx), Q O(d). Moreover, if Y isH an orthonormal basis of d , then so is 7→Y (Q ) . ∈ { α} Hn { α {·} } Proof. Since ∆ is invariant under the rotation group O(d) (writing ∆ = and d d ∇ · ∇ changing variables), if Y n and Q O(d) then Y (Qx) n. That Yα(Qx) is an orthonormal basis of∈ Hd whenever∈ Y (x) is follows from∈ H { } Hn { α } 1 1 Yα(Qx)Yβ (Qx)dσ(x)= Yα(x)Yβ (x)dσ(x)= δα,β, ω Sd−1 ω Sd−1 d Z d Z which holds under a change of variables since dσ is invariant under O(d). d Besides f,g Sd−1 , another useful inner product can be defined on n through h i Nd α α1 αd P the action of differentiation. For α 0, let ∂ := ∂1 ∂d . Let (a)n := a(a + 1) (a + n 1) be the Pochhammer∈ symbol. ··· ··· − Theorem 1.8. For p, q d, define a bilinear form ∈Pn (1.7) p, q := p(∂)q, h i∂ where p(∂) is the differential operator defined by replacing xα in p(x) by ∂α. Then 4 FENG DAI AND YUAN XU d 1. p, q ∂ is an inner product on n; 2.
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