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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 Pthe 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 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.h the reproducingi kernel of this innerP product is k (x, y) := x, y n/n!; that is, n h i k (x, ),p = p(x), p d; h n · i∂ ∀ ∈Pn 3. for p d and q d , ∈Pn ∈ Hn n d p, q =2 p, q Sd−1 . h i∂ 2 h i  n Proof. Let p, q d be given by p(x) = a xα and q(x) = b xα, ∈ Pn |α|=n α |α|=n α where a ,b R. Then, α α ∈ P P (1.8) p, q = a ∂α b xβ = α!a b , h i∂ α β α α |αX|=n |βX|=n |αX|=n which implies, in particular, that p,p > 0 for p = 0. It follows then that , h i∂ 6 h· ·i∂ is an inner product on d. By the multinomial formula, for q (x)= xα, α = n, Pn α | | 1 n ∂β k (x, ), q = xβ yα = q (x), h n · αi∂ n! β ∂yβ α |βX|=n   which shows that kn(x, y) is the reproducing kernel with respect to , ∂ . We now prove item 3. Integrating by parts shows that h· ·i

−kxk2/2 −kxk2/2 ∂if(x)g(x)e dx = f(x)(∂ig(x) xig(x))e dx. Rd − Rd − Z Z Since p(∂)q is a constant, using this integration by parts repeatedly shows that

1 −kxk2/2 p, q ∂ = d/2 p(∂)q(x)e dx h i (2π) Rd Z 1 −kxk2/2 = d/2 q(x)(p(x)+ s(x))e dx (2π) Rd Z where s Πd . Since q d and p d, switching to polar integral and using ∈ n−1 ∈ Hn ∈ Pn the orthogonality of d , we obtain Hn ∞ 1 2n+d−1 −r2/2 ′ ′ ′ p, q ∂ = d/2 r e dr q(x )p(x )dσ(x ). h i (2π) Sd−1 Z0 Z Evaluating the integral in r and simplifying by (1.2) concludes the proof.  A large number of spherical harmonic polynomials can be defined explicitly through differentiation. Let us denote the standard basis of Rd by e = (1, 0, , 0),e = (0, 1, 0,..., 0) ,e = (0, , 0, 1). 1 ··· 2 ··· d ··· Theorem 1.9. Let d> 2. For α Nd, n = α , define ∈ 0 | | ( 1)n (1.9) p (x) := − x 2|α|+d−2∂α x −d+2 . α n d−2 2 ( 2 )n k k k k Then  1. p d and p is the monic spherical harmonic of the form α ∈ Hn α (1.10) p (x)= xα + x 2q (x), q d . α k k α α ∈Pn−2 5

2. pα satisfies the 1 (1.11) p (x)= x p (x) x 2∂ p (x), α+ei i α − 2n + d 2k k i α − 3. p : α = n, α =0or1 is a basis of d . { α | | d } Hn

Proof. Taking the derivative of pα(x) gives immediately the recurrence relation (1.11). Clearly p0(x) = 1. By induction, the recurrence relation shows that pα is a homogeneous polynomial of degree n and it is of the form (1.10). We now show d R that pα is a spherical harmonic. For g n and ρ , a quick computation using d ∈P ∈ i=1 xi∂ig(x)= ng(x) shows that

P(1.12) ∆( x ρg)= ρ(2n + ρ + d 2) x ρ−2g + x ρ∆g. k k − k k k k In particular, setting n = 0 and g(x) = 1 gives ∆( x −d+2) = 0. Furthermore, setting g = p and ρ = 2n d + 2 in (1.12) leads tok k α − − ( 1)n ∆p (x)= − x 2|α|+d−2∂α∆ x −d+2 =0. α 2n(d/2 1) k k {k k } − n d 2 β Thus, pα n. Since x q(x) is a linear combination of the monomials x with β 2, by∈ H (1.10) andk thek linear independence of xα : α = n, α = 0 or1 , d ≥ { | | d } it follows that the elements in the set pα : α = n, αd = 0or 1 are linearly independent. The cardinality of the set is{ | | }

n + d 2 n + d 3 dim d−1 + dim d−1 = − + − , Pn Pn−1 d 2 d 2  −   −  d which is, by a simple identity of binomial coefficients and (1.5), precisely dim n. This completes the proof. H

The right-hand side of (1.9) is called Maxwell’s representation of harmonic poly- nomials ([8, 10]). The complete set of pα : α = n is necessarily linearly depen- dent by its cardinality. Moreover, by (1.9),{ | | }

( 1)n p + ... + p = − x 2|α|+d−2∂α∆ x −d+2 =0, α+2e1 α+2ed n d 2 ( 2 )n k k k k  d which gives dim n−2 linearly dependent relations among pα : α = n . The set P d { | | } pα : α = n evidently contains many bases of n. The basis in item three of Theorem{ | | 1.9 is} but one convenient choice. The proofH of Theorem 1.9 relies on the fact that x −d+2 is a harmonic function in Rd 0 for d> 2. In the case of d = 2, we need tok k replace this function by log x . Since\{ } the case d = 2 corresponds to the classical Fourier series, we leave thek analoguek of Theorem 1.9 for d = 2 to the interested reader. d The basis pα : α = n, αd = 0or1 of n is not orthonormal. In fact, the elements of this{ basis| | are not mutually} orthogonal.H Orthonormal bases can be constructed by applying the Gram-Schmidt process. An explicit orthonormal basis for d will be given in Section 5 in terms of spherical coordinates. Hn 6 FENG DAI AND YUAN XU

2. Projection operators and Zonal harmonics Let L2(Sd−1) denote the space of square integrable functions on Sd−1. Let proj : L2(Sd−1) d n 7→ Hn 2 Sd−1 d d denote the orthogonal projection from L ( ) onto n. If P n then P = P + x 2Q , where P d and Q d , by (1.4),H so that proj∈ P P = P . In n k k n n ∈ Hn n ∈Pn−2 n n particular, (1.11) shows that pα defined in (1.9) is the orthogonal projection of the α function qα(x)= x ; that is, pα = projn qα. This leads to the following: Lemma 2.1. Let p d. Then ∈Pn ⌊n/2⌋ 1 (2.1) proj p = x 2j ∆j p. n 4jj!( n +2 d/2) k k j=0 j X − − α Proof. By linearity, it suffices to consider p being qα(x) = x . By Theorem 1.9, projn qα(x)= pα(x), and the proof amounts to showing that pα(x) defined in (1.9) can be expanded as in (2.1). We use induction on n. The case n = 0 is evident. Assume that (2.1) has been established for m =0, 1,...,n. Applying (2.1) to qα(x), α = n, it follows that | | ∂α x −d+2 =( 1)n2n d 1 x −2n−d+2 k k − 2 − n k k ⌊n/2⌋   1 x 2j∆j xα . × 4jj!( n +2 d/2) k k { } j=0 j X − − Applying ∂i to this identity, we obtain ∂ ∂α x −d+2 = ( 1)n2n d 1 ( 2n d + 2) x −2n−d+2 i k k − 2 − n − − k k ⌊(n+1)/2⌋  1  x 2j [x ∆j xα +2j∆j−1∂ xα ]. × 4jj!( n +1 d/2) k k i { } i{ } j=0 j X − − The terms in the square brackets are exactly ∆j x xα and the constant in front { i } simplifies to ( 1)n+12n+1 d 1 , so that the equation (2.1) holds for p(x) = − 2 − n+1 x xα. This completes the induction.  i  Definition 2.2. The reproducing kernel Z ( , ) of d is uniquely determined by n · · Hn 1 d Sd−1 (2.2) Zn(x, y)p(y)dσ(y)= p(x), p n, x ω Sd−1 ∀ ∈ H ∈ d Z and the requirement that Z (x, ) be an element of d for each fixed x. n · Hn That the kernel is well defined and unique follows from the Riesz representation d theorem applied to the linear functional L(Y ) := Y (x), Y n, for a fixed x Sd−1. ∈ H ∈ Lemma 2.3. In terms of an orthonormal basis Y :1 j dim d of d , { j ≤ ≤ Hn} Hn d dim Hn (2.3) Z (x, y)= Y (x)Y (y), x,y Sd−1, n k k ∈ Xk=1 and, despite (2.3), Z is independent of the particular choice of basis of d . n Hn 7

Proof. Since Z (x, ) d , it can be expressed as Z (x, y) = c Y (y) where n · ∈ Hn n k k k the coefficients are determined by (2.2) as ck = Yk(x). The uniqueness implies that P Zn is independent of the choice of basis. This can also be shown directly as follows. Y d Let n = (Y1, ..., YN ) with N = dim n and regard it as a column vector. Then Y trY ′ H d Zn(x, y) = [ n(x)] n(y). If Yj :1 j N is another orthonormal basis of n, Y′ Y { ≤ ≤ } H then n = Q n. Since the orthonormality of Yj can be expressed as the fact that 1 tr { } d−1 Y (x)[Y (x)] dσ(x) is an identity matrix, it follows readily that Q is an ωd S n n orthogonal matrix. Hence, Z (x, y) = [Y (x)]trQtrQY (y) = [Y′ (x)]trY′ (y).  R n n n n n The reproducing kernel is also the kernel for the projection operator. Lemma 2.4. The projection operator can be written as 1 (2.4) projn f(x)= f(y)Zn(x, y)dσ(y). ω Sd−1 d Z d Proof. Since projn f n, it can be expanded in terms of the orthonormal basis ∈ H d d Yj, 1 j Nn , Nn = dim n, of n, where the coefficients are determined by {the orthonormality,≤ ≤ } H H

N n 1 projn f(x)= cjYj (x) with cj = f(y)Yj (y)dσ(y). ω Sd−1 j=1 d X Z If we pull out the integral in front of the sum, this is (2.4) by (2.3). 

Lemma 2.5. The kernel Z ( , ) satisfies the following properties: n · · 1. For every ξ, η Sd−1, ∈ 1 (2.5) Zn(ξ,y)Zn(η,y)dσ(y)= Zn(ξ, η). ω Sd−1 d Z 2. Z (x, y) depends only on x, y . n h i Proof. By Corollary 1.7, the uniqueness of Zn(x, y) shows that Zn(Qx,Qy) = Z (x, y) for all Q O(d). Since for x, y Sd−1 there exists a Q SO(d) such that n ∈ ∈ ∈ Qx = (0, ..., 0, 1) and Qy = (0, ..., 0, 1 x, y 2, x, y ), this shows that Z (x, y) − h i h i n depends only on x, y . q  h i

From the second property of the lemma, Zn(x, y) = Fn( x, y ), which is often called a zonal harmonic, since it is harmonic and depends onlyh oni x, y . We now h i derive a closed formula for Fn, which turns out to be a multiple of the Gegenbauer polynomial, Cλ, of degree n defined, for λ> 0 and n N , by n ∈ 0 n n 1−n λ (λ)n2 n 2 , 2 1 (2.6) C (x) := x 2F1 − ; , n n! 1 n λ x2  − −  where 2F1 is the . The properties of the Gegenbauer poly- nomials are collected in Appendix B. Theorem 2.6. For n N and x, y Sd−1, d 3, ∈ 0 ∈ ≥ n + λ d 2 (2.7) Z (x, y)= Cλ( x, y ), λ = − . n λ n h i 2 8 FENG DAI AND YUAN XU

d Proof. Let p n. By Theorem 1.8, p(x) = kn(x, ),p ∂ . For fixed x, it follows from the same∈ theorem H that h · i p(x)= k (x, ),p = proj (k (x, )),p h n · i∂ h n n · i∂ n 2 (d/2)n = projn[kn(x, )](y)p(y)dσ(y). ω Sd−1 · d Z Since the kernel Zn( , ) is uniquely determined by the reproducing property, this · · n d shows that Zn(x, y)=2 ( 2 )n projn[kn(x, )](y). Since kn(x, ) is a homogeneous · · j polynomial of degree n and, taking the derivative on y, we have ∆ kn(x, y) = x 2jk (x, y), as is easily seen from ∂ k (x, y)= x k (x, y), Lemma 2.1 shows, k k n−2j i n i n−1 for x, y Sd−1, that ∈ ⌊n/2⌋ d n−2j n d ( 2 )n2 Zn(x, y)=2 projn[kn(x, )](y)= kn−2j (x, y). 2 n · j!(1 n λ) j=0 j   X − − Using the fact 1/(n 2j)! = ( n) /n!=22j ( n ) ( −n+1 ) /n!, we conclude then − − 2j − 2 j 2 j n ⌊n/2⌋ n −n+1 n + λ (λ) 2 ( )j ( )j Z (x, y)= n − 2 2 x, y n−2j n λ n! j!(1 n λ) h i j=0 j X − − n n 1−n n + λ (λ)n2 n 2 , 2 1 = x, y 2F1 − ; 2 , λ n! h i 1 n λ x, y ! − − h i from which the stated result follows from (2.6). 

d d Let Yi : 1 i dim n be an orthonormal basis of n. Then (2.7) states that { ≤ ≤ H } H

dim Hd n n + λ d 2 (2.8) Y (x)Y (y)= Cλ( x, y ), λ = − . j j λ n h i 2 j=1 X This identity is usually referred to as the addition formula of spherical harmonics, since for d = 2 it is the addition formula of the cosine function (see Section 6). Corollary 2.7. For n N and x, y Sd−1, d 3, ∈ 0 ∈ ≥ (2.9) Z (x, y) dim d and Z (x, x) = dim d . | n |≤ Hn n Hn n+λ λ Proof. Set Fn(t) := λ Cn (t). By (2.7), Zn(x, x) = Fn(1) is a constant for all x Sd−1. Setting x = y in (2.3) and integrating over Sd−1, we obtain ∈ d dim Hn 1 1 2 d Fn(1) = Zn( x, x )dσ(x)= Yk (x)dσ(x) = dim n. ωd Sd−1 h i ωd Sd−1 H Z Z kX=1 The inequality follows from applying the Cauchy-Schwarz inequality to (2.3). 

d−2 Because of the relation (2.7), the Gegenbauer polynomials with λ = 2 are also called ultraspherical polynomials. A number of properties of the Gegenbauer polynomials can be obtained from the zonal spherical harmonics. For example, the corollary implies that Cλ(1) = λ dim d . Here is another example: n n+λ Hn 9

d−2 λ Corollary 2.8. For λ = 2 , the Gegenbauer polynomials Cn satisfy the orthogo- nality relation 1 ωd−1 2 1 (2.10) Cλ(t)Cλ (t)(1 t )λ− 2 dt = hλδ , ω n m − n m,n d Z−1 where λ hλ = Cλ(1). n n + λ n Proof. Set again F (t)= n+λ Cλ(t). Since Z (x, ) and Z (x, ) are orthogonal over n λ n n · m · Sd−1, and by (A.5.1), their integrals can be written as an integral of one variable, we obtain

1 − ωd−1 2 d 3 1 Fn(t)Fm(t)(1 t ) 2 dt = Fn( x, y )Fm( x, y )dσ(y) ω − ω Sd−1 h i h i d Z−1 d Z = F (1)δ = (dim d )δ , n m,n Hn m,n where the second line follows from (2.5) and Corollary 2.7. 

The functions on Sd−1 that depend only on x, y are analogues of radial functions on Rd . For such functions, there is a Funk-Heckeh i formula given below. Theorem 2.9. Let f be an integrable function such that 1 f(t) (1 t2)(d−3)/2dt −1 | | − is finite and d 2. Then for every Y d , ≥ n ∈ Hn R d−1 (2.11) f( x, y )Yn(y)dσ(y)= λn(f)Yn(x), x S , Sd−1 h i ∈ Z where λn(f) is a constant defined by

d−2 1 2 Cn (t) d−3 2 2 λn(f)= ωd−1 f(t) d−2 (1 t ) dt. 2 − Z−1 Cn (1) Proof. If f is a polynomial of degree m, then we can expand f in terms of the Gegenbauer polynomials m d−2 d−2 k + 2 2 f(t)= λk d−2 Ck (t). 2 Xk=0 where λk are determined by the orthogonality of the Gegenbauer polynomials,

1 −2 cd d d−3 2 2 2 λk = d−2 f(t)Ck (t)(1 t ) dt, 2 −1 − Ck (1) Z − −1 1 2 d 3 and c = (1 t ) 2 dt = ω /ω . From (2.7) and the reproducing property d −1 − d d−1 of Z (x, y), it follows that for n m n R ≤ 1 d−1 f( x, y )Yn(y)dσ(y)= λnYn(x), x S . ω Sd−1 h i ∈ d Z Since λn/ωd = λn(f) by definition, we have established the Funk-Hecke formula (2.11) for polynomials, and hence, by the Weierstrass theorem, for continuous func- tions, and the function satisfying the integrable condition in the statement can be approximated by a sequence of continuous functions.  10 FENG DAI AND YUAN XU

3. Zonal basis of spherical harmonics In view of the addition formula of spherical harmonics, one may ask whether there is a basis of spherical harmonics that consists entirely of zonal harmonics. This question is closely related to the problem of interpolation on the sphere. d Throughout this subsection, we fix n, set N = dim n and fix Y1,...,YN as an orthonormal basis for d . Let x ,...,x be a collectionH of{ points on Sd}−1. Hn { 1 N } We let M1 := Y1(x1) and, for k =2, 3,...,N, define matrices

Y1(x) Y1(x1) ... Y1(xk) . . .  Mk−1 .  Mk :=  . ... .  ,Mk(x) := . Y (x) Y (x ) ... Y (x )  k−1   k 1 k k   Y (x ),...Y (x ) Y (x),     k 1 k k−1 k  T   The product of MN and its transpose MN can be summed on applying the addition T N formula (2.8), as MN MN = [Zn(xi, xj )]i,j=1, which shows, in particular, (3.1) det[Z (x , x )]N = (det M )2 0. n i j i,j=1 N ≥ This motivates the following definition. d−1 Definition 3.1. A collection of points x1,...,xN in S is called a fundamental system of degree n on the sphere Sd−1 {if } N det Cλ( x , x ) > 0, λ = d−2 . n h i j i i,j=1 2 Lemma 3.2. There exists a fundamental system of degree n on the sphere. Proof. The existence of a fundamental system follows from the linear independence of Y ,...,Y . Indeed, we can clearly choose x Sd−1 such that det M = { 1 N } 1 ∈ 1 Y1(x1) = 0. Assume that x1,...,xk, 1 k N 1, have been chosen such that det M 6 = 0. The determinant det M ≤(x)≤ is a− polynomial of x and cannot be k 6 k+1 identically zero by the linear independence of Y1,...,Yk+1 , so that there is a d−1 { } xk+1 S such that det Mk+1 = det Mk(xk+1) = 0. In this way, we end up with ∈ d−1 6 a collection of points x1,...,xN on S that satisfies det MN = 0, which implies N { } 6 det Cλ( x , x ) = c (det M )2 > 0, where c = λN /(n + λ)N .  n h i j i i,j=1 N N N The proof shows, in fact, that there are infinitely many fundamental systems. Indeed, regarding x1,...,xN as variables, we say that det MN is a (d 1)N- dimensional polynomial in these variables, and its zero set is an algebraic− surface of R(d−1)N , which necessarily has measure zero.

Theorem 3.3. If x1,...,xN is a fundamental system of points on the sphere, λ { } d−2 d then C ( , x ): i =1, 2,...,N , λ = , is a basis of Sd−1 . { n h· ii } 2 Hn| Proof. Let P (x)= n+λ Cλ( , x ). The addition theorem (2.8) gives i λ n h· ii N Pi(x)= Yk(xi)Yk(x), i =1, 2,...,N, kX=1 which shows that P1,...,PN is expressed in the basis Y1,...,YN with transition matrix given by M{ = [Y (x )]} N . Since x ,...,x { is fundamental,} the matrix N k i k,i=1 { 1 N } is invertible, by (3.1). We can then invert the system to express Yk as a linear combination of P1,...,PN , which completes the proof.  11

λ Sd−1 A word of caution is in order. The polynomial Cn ( x, xi ) is, for x , a linear combination of the spherical harmonics accordingh to thei addition∈ formula. It is not, however, a homogeneous polynomial of degree n in x Rd; rather, it is n λ ∈ the restriction of the homogeneous polynomial x Cn ( x/ x ,y ) to the sphere. k k d h k kd i This is a situation in which the distinction between n and n Sd−1 is called for; see the discussion below Definition 1.1. H H | Fundamental sets of points are closely related to the problem of interpolation. Indeed, it can be stated as follows: for a given set of data (x ,y ):1 j N , { j j ≤ ≤ } x Sd−1 and y R, there is a unique element Y d such that Y (x ) = y , j ∈ j ∈ ∈ Hn j j j = 1,...,N, if and only if the points x1,...,xN form a fundamental system on the sphere. Much more interesting and{ challenging} is the problem of choosing points in such a way that the resulting basis of zonal spherical harmonics has a relatively simple structure. A related result is a zonal basis for the space n of homogeneous polynomials of d P degree n and the space Πn of all polynomials of degree at most n. Theorem 3.4. There exist points ξ Sd−1, 1 j rd := dim d, such that j,n ∈ ≤ ≤ n Pn (i) x, ξ n :1 j rd is a basis for d of degree n. {h j,ni ≤ ≤ n} Pn (ii) For each polynomial f Πd , there exist polynomials p :[ 1, 1] R for ∈ n j − 7→ 1 j rd such that ≤ ≤ n d rn f(x)= p ( x, ξ ). j h j,ni j=1 X Proof. Following the proof of the existence of fundamental system of points, it is easy to see that there exist points ξj,n such that f(ξj,n) = 0 if and only if f = 0 for all f d. For the proof of (i), we first deduce by the binomial formula that ∈Pn n! f (x) := x, ξ n = ξα xα, 1 j rd . j,n h j,ni α! j,n ≤ ≤ n |αX|=n Let f d. Then f(x)= a xα. By (1.8), ∈Pn |α|=n α f,fP = n! a ξα = n!f(ξ ), h j,ni∂ α j,n j,n |αX|=n which implies, by the choice of ξ , that f,f =0, 1 j rd , if and only if j,n h j,ni∂ ≤ ≤ n f = 0. Thus, f ⊥ = 0 . Since f d. This proves (i). { j,n} { } j,n ∈Pn For the proof of (ii), let m be an integer, 0 m n 1, and let fj,m,n := m d ≤ ≤ − x, ξj,n . For f m, it follows as above that f,fj,m,n ∂ = m!f(ξj,m), 1 j hd i ∈ P d h i ≤ ≤ rn. Hence, if f,fj,m,n ∂ = 0 for 1 j rn, then f(ξj,m) = (fg)(ξj,m) = 0 for d h i d ≤ ≤ 1 j rn and all g n−m, which implies by the choice of ξj,n and the fact that ≤ ≤d ∈P d d fg n, that fg =0 or f = 0. Consequently, m = span fj,m,n :0 j rn for ∈P n P { ≤ ≤ } 0 m n. Since Πd = d , this proves (ii).  ≤ ≤ n m=0 Pm 4.PLaplace-Beltrami operator The operator in the section heading is the spherical part of the Laplace operator, which we denote by ∆0. The operator ∆0 plays an important role for analysis on the sphere. The usual approach to deriving this operator relies on an expression of the Laplace operator under a change of variables, which we describe first. 12 FENG DAI AND YUAN XU

For x Rd let x u = u(x) be a change of variables that is a bijection, so that we can also∈ write x7→= x(u). Introduce the tensors

d d ∂xk ∂xk i,j ∂ui ∂uj gi,j := and g := , 1 i, j d, ∂ui ∂uj ∂xk ∂xk ≤ ≤ kX=1 kX=1 d −1 i,j and let g := det(gi,j )i,j=1. Then (gi,j ) = (g ). A general result in Riemannian geometry, or a bit of tensor analysis, shows that the Laplace operator satisfies

d d d ∂2 1 ∂ ∂ (4.1) ∆ = = √ggi,j . ∂x2 g ∂u ∂u i=1 i √ i=1 j=1 i j X X X The Laplace–Beltrami operator, i.e., the spherical part of the Laplace operator, can then be derived from (4.1) by the change of variables x (r, ξ1,...,ξd−1), where d−1 7→ r> 0 and ξ = (ξ1,...,ξd) S . For this approach and a derivation of (4.1), see [10]. We shall adopt an approach∈ that is elementary and self-contained. Lemma 4.1. In the spherical–polar coordinates x = rξ, r > 0, ξ Sd−1, the Laplace operator satisfies ∈ ∂2 d 1 ∂ 1 (4.2) ∆ = + − + ∆ , ∂r2 r ∂r r2 0 where d−1 d−1 d−1 d−1 ∂2 ∂2 ∂ (4.3) ∆ = ξ ξ (d 1) ξ . 0 ∂ξ2 − i j ∂ξ ∂ξ − − i ∂ξ i=1 i i=1 j=1 i j i=1 i X X X X Proof. Since ξ Sd−1, we have ξ2 + ... + ξ2 = 1. We evaluate the Laplacian ∆ ∈ 1 d under a change of variables (x1,...,xd) (r, ξ1,...,ξd−1) under x = rξ, which has inverse ξ = x / x ,...,ξ = x /7→x , r = x . The chain rule leads to 1 1 k k d−1 d−1 k k k k d−1 ∂ 1 ∂ ξ ∂ ∂ = i ξ + ξ , 1 i d 1, ∂x r ∂ξ − r j ∂ξ i ∂r ≤ ≤ − i i j=1 j (4.4) X ∂ x d−1 ∂ x ∂ = d ξ + d . ∂x − r2 j ∂ξ r ∂r d j=1 j X If we apply the product rule for the partial derivative on xd, it follows that d−1 d−1 1 ∂ ξ ∂ ∂ 2 ∆= i ξ + ξ r ∂ξ − r j ∂ξ i ∂r i=1 i j=1 j X  X  1 d−1 ∂ ∂ 2 1 d−1 ∂ 1 ∂ + ξ2 ξ + + (1 ξ2) ξ + , d − r j ∂ξ ∂r − d −r2 j ∂ξ r ∂r j=1 j j=1 j  X   X  where we have used xd = rξd, from which a straightforward, though tedious, com- 2 2  putation, and simplification using ξ1 + ... + ξd = 1, establishes (4.2) and (4.3). The Laplace–Beltrami operator also satisfies a recurrence relation that can be d−1 used to derive an explicit formula for ∆0 under a given coordinate system of S . We write ∆0,d instead of ∆0 when we need to emphasize the dimension. 13

Lemma 4.2. Let ∆ be the Laplacian–Beltrami operator for Sd−1. For ξ Sd−1, 0,d ∈ write ξ = (√1 t2η,t) with 1 t 1 and η Sd−2. Then − − ≤ ≤ ∈ 1 ∂ d−1 ∂ 1 2 2 (4.5) ∆0,d = d−3 (1 t ) + 2 ∆0,d−1. (1 t2) 2 ∂t − ∂t 1 t −   − Proof. We work with the expression for ∆0,d in (4.3) and make a change of variables (ξ ,...,ξ ) (η ,...,η ,t) defined by 1 d−1 7→ 1 d−2 ξ = 1 t2η ,...,ξ = 1 t2η , ξ = t, 1 − 1 d−2 − d−2 d−1 where we have switchedp ξd−1 and ξd for convenience.p The chain rule gives ∂ 1 ∂ ∂ t d−2 ∂ ∂ = , 1 i d 2, and = 2 ηj + , ∂ξ √ 2 ∂η ≤ ≤ − ∂ξ 1 1 t ∂η ∂t i 1 t i d− j=1 j − − X which can be used iteratively to compute ∆0,d on writing (4.3) as

2 d−2 2 d−2 2 2 ∂ ∂ ∂ ∆0,d = (1 t ) 2 + 2 2t ξi − ∂ξ ∂ξ − ∂ξ ∂ξ 1 d−1 i=1 i i=1 i d− X X d−2 d−2 d−1 ∂2 ∂ ξ ξ (d 1) . − i j ∂ξ ∂ξ − − ∂ξ i=1 j=1 i j i=1 i X X X A straightforward computation and another use of (4.3) then leads to ∂2 ∂ 1 ∆ = (1 t2) (d 1)t + ∆ , 0,d − ∂t2 − − ∂t 1 t2 0,d−1 − which is precisely (4.5). 

The formula (4.3) gives an explicit expression for ∆0 in the local coordinates of d−1 S . An explicit formula for ∆0 in terms of spherical coordinates will be given in Section 5. Let = (∂ , . . . , ∂ ). The proof of Lemma 4.1 also shows that ∇ 1 d 1 ∂ (4.6) = + ξ , x = rξ, ξ Sd−1, ∇ r ∇0 ∂r ∈ where 0 is the spherical gradient, which is the spherical part of and involves only derivatives∇ in ξ. Its explicit expression can be read off from (4.4).∇ We shall not need this expression and will be content with the following expression. Corollary 4.3. Let f C2(Sd−1). Define F (y) := f(y/ y ), y Rd. Then ∈ k k ∈ (4.7) ∆ f(x) = ∆F (x) and f(x)= F (x), x Sd−1. 0 ∇0 ∇ ∈ The corollary follows immediately from (4.2) and (4.6), since x/ x is indepen- k k dent of r. The expressions in (4.7) show that ∆0 and 0 are independent of the d−1 ∇ coordinates of S . In fact, we could take (4.7) as the definition of ∆0 and 0. The usual Laplacian ∆ can be expressed in terms of the dot product ∇of , ∆= , which can also be written–as is often in physics textbooks–as ∆ = ∇2. The analogue∇ · ∇ of this identity also holds on the sphere. ∇ Lemma 4.4. The Laplace–Beltrami operator satisfies (4.8) ∆ = . 0 ∇0 · ∇0 14 FENG DAI AND YUAN XU

Proof. An application of (4.6) gives immediately 1 1 ∂ ∂ 1 ∂2 ∆= = + ξ + ξ + . ∇ · ∇ r2 ∇0 · ∇0 r ∇0 ∂r ∂r r ∇0 ∂r2     d−1 We note that ξ 0f(ξ) = 0, since ξ S is in the normal direction of ξ whereas f(ξ), by (4.6),· ∇ is on the tangent plane∈ at ξ. Hence, we see that ∇0 ∂ 1 1 1 ∂ ξ = ξ + ξ =0. ∂r r ∇0 −r2 · ∇0 r · ∇0 ∂r   Using (4.6), a quick computation gives ξ = d 1, so that by the product rule, ∇0 · − ∂ ∂ ∂ ∂ ξ = ξ + ξ = (d 1) . ∇0 ∂r ∇0 · ∂r ∂r · ∇0 − ∂r   Consequently, we conclude that ∂2 d 1 ∂ 1 ∆= + − + . ∂r2 r ∂r r2 ∇0 · ∇0 Comparing this with (4.2) completes the proof.  Our next result shows that the spherical harmonics are eigenfunctions of the Laplace-Beltrami operator.

Theorem 4.5. The spherical harmonics are eigenfunctions of ∆0, (4.9) ∆ Y (ξ)= n(n + d 2)Y (ξ), Y d , ξ Sd−1. 0 − − ∀ ∈ Hn ∈ Sd−1 d n Proof. Let x = rξ, ξ . Since Y n is homogeneous, Y (x) = r Y (ξ) and by (4.2), ∈ ∈ H 0 = ∆Y (x)= n(n 1)rn−2Y (ξ) + (d 1)nrn−2Y (ξ)+ rn−2∆ Y (ξ), − − 0 which is (4.9) upon dividing by rn−2. 

The identity (4.9) also implies that ∆0 is self-adjoint, which can also be proved directly and will be treated in the last section of this chapter, together with a number of other properties of the Laplace-Beltrami operator.

5. Spherical harmonics in spherical coordinates

The polar coordinates (x1, x2) = (r cos θ, r sin θ), r 0, 0 θ 2π, give coordinates for S1 when r = 1. The high-dimensional analogue≥ ≤ is the≤ spherical polar coordinates defined by

x1 = r sin θd−1 ... sin θ2 sin θ1,  x2 = r sin θd−1 ... sin θ2 cos θ1,  (5.1)   ··· xd−1 = r sin θd−1 cos θd−2,  xd = r cos θd−1,  where r 0, 0 θ 2π,0 θ π for i =2,...,d 1. When r = 1 these are the 1  i coordinates≥ for≤ the unit≤ sphere≤ Sd≤−1, and they are in− fact defined recursively by x = (ξ sin θ , cos θ ) Sd−1, ξ Sd−2. d−1 d−1 ∈ ∈ d−1 Let dσ = dσd be Lebesgue measure on S . Then it is easy to verify that d−2 (5.2) dσd(x) = (sin θd−1) dθd−1dσd−1(ξ). 15

1 Since the Lebesgue measure of S is dθ1, it follows by induction that d−2 d−j−1 (5.3) dσ = dσd = (sin θd−j) dθd−1 ...dθ2dθ1 j=1 Y in the spherical coordinates (5.1). Furthermore, (5.2) shows that π d−2 (5.4) f(x)dσd(x)= f(ξ sin θ, cos θ)dσd−1(ξ)(sin θ) dθ. Sd−1 Sd−2 Z Z0 Z The orthogonality (2.10) of the Gegenbauer polynomials can be written as π d−1 λ λ d−2 √πΓ( 2 ) λ d−2 (5.5) C (cos θ)C (cos θ)(sin θ) dθ = h δm,n, λ = . n m Γ( d ) n 2 Z0 2 Together with (5.4), this allows us to write down a basis of spherical harmonics in terms of the Gegenbauer polynomials in the spherical coordinates. Theorem 5.1. For d> 2 and α Nd, define ∈ 0 d−2 j+1 −1 |α| |α | λj (5.6) Yα(x) := [hα] r gα(θ1) (sin θd−j) Cαj (cos θd−j), j=1 Y j where gα(θ1) = cos αd−1θ1 for αd =0, sin αd−1θ1 for αd =1, α = αj +...+αd−1, λ = αj+1 + (d j 1)/2, and | | j | | − − d−2 d−j+1 2 αj !( 2 )|αj+1|(αj + λj ) [hα] := bα d−j j+1 j=1 (2λj )αj ( 2 )|α |λj Y in which bα =2 if αd−1 +αd > 0, while bα =1 otherwise. Then Yα : α = n, αd = d { | | 0, 1 is an orthonormal basis of ; that is, Y , Y Sd−1 = d . } Hn h α βi α,β Proof. To see that Yα is a homogeneous polynomial, we use, by (5.1), the relation cos θ = x / x2 + ... + x2 for 1 k d 1 to rewrite (5.6) as k k+1 1 k+1 ≤ ≤ − q d−2 −1 2 2 αj /2 λj xd−j+1 Yα(x) = [hα] g(x) (x1 + ... + xd−j+1) Cα j 2 2 j=1 x + ... + x Y  1 d−j+1  αd−1 αd−1 q where g(x) = ρ cos αd−1θ1 for αd = 0, ρ cos αd−1θ1 for αd = 1, with 2 2 ρ = x1 + x2. Since x1 = ρ sin θ1 and x2 = ρ cos θ1 by (5.1), g(x) is either the real αd−1 part or the imaginary part of (x2 + ix1) , which shows that it is a homogeneous p λ polynomial of degree αd−1 in x. Since Cn (t) is even when n is even and odd when n is odd, we see that Y d. Using (5.4), we see that α ∈Pn −1 −1 2π hα hα′ Y , Y ′ Sd−1 = g (θ )g ′ (θ )dθ h α α i ω α 1 α 1 1 d Z0 d−2 π λj λj 2λj ′ Cαj (cos θd−j )Cα (cos θd−j)(sin θd−j) dθd−j × j j=1 0 Y Z from which the orthogonality follows from the orthogonality of the Gegenbauer polynomials (5.4) and that of cos mθ and sin mθ on [0, 2π), and the formula for hα follows from the normalization constant of the Gegenbauer polynomial.  16 FENG DAI AND YUAN XU

For d = 2 and the polar coordinates (x1, x2) = (r cos θ, r sin θ), it is easy to see that 0 = ∂θ, where ∂θ = ∂/∂θ. Hence by (4.8), the Laplace-Beltrami operator ∇ 2 for d = 2 is ∆0 = ∂θ . Using (4.5) iteratively, we see that the Laplace-Beltrami operator ∆0 has an explicit formula in the spherical coordinates (5.1),

1 ∂ d−2 ∂ (5.7) ∆0 = d−2 sin θd−1 sin θ ∂θd−1 ∂θd−1 d−1   d−2 1 ∂ ∂ + sinj−1 θ . 2 2 j−1 ∂θ j ∂θ j=1 sin θd−1 sin θj+1 sin θj j j X ···   6. Spherical harmonics in two and three variables Since spherical harmonics in two and three variables are used most often in applications, we state their properties in this section.

2 6.1. Spherical harmonics in two variables. For d = 2, dim n = 2. An or- 2 H n thogonal basis of n is given by the real and imaginary parts of (x1 + ix2) , since both are homogeneousH of degree n and are harmonic as the real and imaginary 2 parts of an analytic function. In polar coordinates (x1, x2) = (r cos θ, r sin θ) of R , this basis is given by (1) n (2) n (6.1) Yn (x)= r cos nθ, Yn (x)= r sin nθ. Hence, restricting to the circle S1, the spherical harmonics are precisely the cosine and sine functions. In particular, spherical harmonic expansions on S1 are the classical Fourier expansions in cosine and sine functions. As homogeneous polynomials, the basis (6.1) is given explicitly in terms of the Tn and Un defined by sin(n + 1)θ T (t) = cos nθ and U (t)= , where t = cos θ, n n sin θ 1 which are related to the Gegenbauer polynomials: Un(t)= Cn(t) and

1 λ 2 lim Cn (x)= Tn(x). λ→0+ λ n The basis in (6.1) can be rewritten then as x x (6.2) Y (1)(x)= rnT 1 , Y (2)(x)= rn−1x U 1 , n n r n 2 n−1 r     2 2 which shows explicitly that these are homogeneous polynomials since r = x1 + x2 and both T (t) and U (t) are even if n is even, odd if n is odd. n n p When d = 2, the zonal polynomial is given by Tn( x, y ) = cos n(θ φ) and the addition formula (2.8) becomes, by (6.1), the additionh formulai − cos nθ cos nφ + sin nθ sin nφ = cos n(θ φ). − The expression (4.2) of the Laplace operator in polar coordinates becomes d2 1 d 1 d2 ∆= + + ; dr r dr r2 dθ2 1 2 2 in particular, the Laplace–Beltrami operator on S is simply ∆0 = d /dθ . 17

3 6.2. Spherical harmonics in three variables. The space n of spherical har- monics of degree n has dimension 2n +1. For d = 3 the sphericalH polar coordinates (5.1) are written as

x1 = r sin θ sin φ, (6.3) x = r sin θ cos φ, , 0 θ π, 0 φ< 2π, r> 0.  2 ≤ ≤ ≤ x3 = r cos θ, S2 Sd−1 The surface area of is 4π and the integral over is parameterized by π 2π (6.4) f(x)dσ = f(sin θ sin φ, sin θ cos φ, cos θ)dφ sin θdθ. S2 Z Z0 Z0 The orthogonal basis (5.6) in spherical coordinates becomes 1 n k k+ 2 Yk,1(θ, φ) = (sin θ) C (cos θ)cos kφ, 0 k n, (6.5) n−k ≤ ≤ k+ 1 Y n (θ, φ) = (sin θ)kC 2 (cos θ) sin kφ, 1 k n. k,2 n−k ≤ ≤ Their L2(S2) norms can be deduced from (5.6). This basis is often written in terms k of the associated Pn (t) defined by dk P k(x) := ( 1)n(1 x2)k/2 P (x)=(2k 1)!!( 1)n(1 x2)k/2Ck+1/2(x), n − − dxk n − − − n−k 0 where Pn(t)= Cn(t) denotes the Legendre polynomial of degree n (see Appendix B k ikφ −ikφ for properties of Pn and Pn ), and in terms of e ,e instead of cos kφ, sin kφ . In this way an orthonormal basis of 3 is given{ by } { } Hn (2n + 1)(n k)! 1/2 (6.6) Y (θ, φ)= − P k(cos θ)eikφ, n k n. k,n (n + k)! n − ≤ ≤   The addition formula (2.8) then reads, assuming that x and y have spherical coor- dinates (θ, φ) and (θ′, φ′), respectively, n (6.7) Y (θ, φ)Y (θ′, φ′)=(2n + 1)P ( x, y ). k,n k,n n h i kX=−n In terms of the coordinates (6.3), the Laplace–Beltrami operator is given by 1 ∂ ∂ 1 ∂2 (6.8) ∆0 = sin θ + sin θ ∂θ ∂θ sin2 θ ∂φ2   as seen from (5.7).

7. Representation of the rotation group In this section we show that the representation of the group SO(d) in spaces of harmonic polynomials is irreducible. A representation of SO(d) is a homomorphism from SO(d) to the group of nonsingular continuous linear transformations of L2(Sd−1). We associate with each element Q SO(d) an operator T (Q) in the space of L2(Sd−1), defined by ∈ (7.1) T (Q)f(x)= f(Q−1x), x Sd−1. ∈ Evidently, for each Q SO(d), T (Q) is a nonsingular linear transformation of L2(Sd−1) and T is a homomorphism,∈ T (Q Q )= T (Q )T (Q ), Q ,Q SO(d). 1 2 1 2 ∀ 1 2 ∈ 18 FENG DAI AND YUAN XU

Thus, T is a representation of SO(d). Since dσ is invariant under rotations, 2 d−1 T (Q)f 2 = f 2 in the L (S ) norm, so that T (Q) is unitary. k A lineark spacek k is called invariant under T if T (Q) maps to itself for all Q SO(d). The nullU space and L2(Sd−1) itself are trivial invariantU subspaces. A representation∈ T is called irreducible if it has only trivial invariant subspaces. The space d is an invariant subspace of T in (7.1). Hn Let Tn,d denote the representation of SO(d) corresponding to T in the invariant subspace d . We want to show that T is irreducible. Hn n,d Lemma 7.1. A spherical harmonic Y d is invariant under all rotations in ∈ Hn SO(d) that leaves xd fixed if and only if x d 2 (7.2) Y (x)= c x nCλ d , λ = − . k k n x 2 k k where c is a constant.

Proof. If Y is invariant under rotations that fix xd and is a homogeneous polynomial of degree n, then it can be written as

Y (x)= b xn−2j (x2 + ... + x2 )j = c xn−2j x 2j. j d 1 d−1 j d k k 0≤Xj≤n/2 0≤Xj≤n/2 where the second equal sign follows from expanding ( x 2 x2)j and changing the k k − d order of summation. Since Y is harmonic, computing ∆Y (x) = 0 shows that cj satisfies the recurrence relation 4(j + 1)(n j 1)c + (n 2j)(n 2j 1)c =0. − − j+1 − − − j Solving the recurrence equation for cj we conclude that

n 1−n ( 2 )j ( 2 )j n−2j 2j Y (x)= c0 − xd x . j!(1 n d 2)j k k 0≤Xj≤n/2 − − − Consequently, (7.2) follows from the formula (??) for Gegenbauer polynomials. Since the function Y in (7.2) is clearly invariant under all rotations that fix xd and we have just shown that it is harmonic, the proof is complete. 

Theorem 7.2. The representation T of SO(d) on d is irreducible. n,d Hn d Proof. Assume that is an invariant subspace of n and is not a null space. U d H U Let Yj : 1 j M , M dim n, be an orthonormal basis of . Follow- ing the{ proof≤ of Lemma≤ } 2.5, there≤ isH a polynomial F (t) of one variableU such that M j=1 Yj (x)Yj (y)= F ( x, y ). In particular, setting y = ed = (0, ..., 0, 1) shows that d h i F ( x, ed ) is in n and it is evidently invariant under rotations in SO(d) that fix xd. Ph i H x Hence, by Lemma 7.1 F ( x, e ) = c x nCλ( d ). In particular, this shows that h di k k n kxk n λ xd ⊥ x Cn ( kxk ) . On the other hand, let denote the orthogonal complement of k k d ∈U ⊥ U −1 in n. If f and g , then T (Q)f,g Sd−1 = f,T (Q )g Sd−1 = 0, which U H ⊥∈U ∈U h i d h i shows that is also an invariant subspace of n. Applying the same argument U x H as for shows then x nCλ( d ) ⊥, which contradicts to ⊥ = 0 . Thus, U k k n kxk ∈U U∩U { } must be trivial.  U 19

8. Angular derivatives and the Laplace–Beltrami operator

Consider the case d = 2 and the polar coordinates (x1, x2) = (r cos θ, r sin θ). Let ∂r = ∂/∂r and ∂θ = ∂/∂θ, while we retain ∂1 and ∂2 for the partial derivatives with respect to x1 and x2. It then follows that sin θ cos θ ∂ = cos θ∂ ∂ , ∂ = sin θ∂ + ∂ . 1 r − r θ 2 r r θ From these relations it follows easily that the angular derivative ∂θ can also be 2 written as ∂θ = x1∂2 x2∂1 and the operator ∆0 is ∆0 = ∂θ . We introduce angular derivatives in higher− dimensions as follows. Definition 8.1. For x Rd and 1 i = j d, define ∈ ≤ 6 ≤ ∂ (8.1) Di,j := xi∂j xj ∂i = , − ∂θi,j where θi,j is the angle of polar coordinates in the (xi, xj )-plane, defined by (xi, xj )= r (cos θ , sin θ ), r 0 and 0 θ 2π. i,j i,j i,j i,j ≥ ≤ i,j ≤ d d By its definition with partial derivatives on R , Di,j acts on R , yet the second equality in (8.1) shows that it acts on the sphere Sd−1. Thus, for f defined on Rd, (8.2) (D f)(ξ)= D [f(ξ)], ξ Sd−1, i,j i,j ∈ where the right-hand side means that Di,j is acting on f(ξ). d Since Dj,i = Di,j, the number of distinct operators Di,j is 2 . The operator ∆ can be decomposed− in terms of them. 0  d−1 Theorem 8.2. On S , ∆0 satisfies the decomposition 2 (8.3) ∆0 = Di,j . 1≤Xi

Let Qi,j,θ denote a rotation by the angle θ in the (xi, xj )-plane, oriented so that (xi, xj ) = (s cos θ,s sin θ). Then T (Qi,j,θ), defined in (7.1), maps f into T (Qi,j,θ)f(x) = f(Qi,j,−θx). Written explicitly, for example, for (i, j) = (1, 2), we have (8.4) T (Q f)(x)= f(x cos θ + x sin θ, x sin θ + x cos θ, x ,...,x ). 1,2,θ 1 2 − 1 2 3 d Then Di,j is the infinitesimal operator of T (Q),

dT (Qi,j,θ) ∂ ∂ (8.5) = xi xj = Di,j , dθ θ=0 ∂xj − ∂xi

where the first equality follows from (8.4). The infinitesimal operator plays an important role in representation theory; see, for example, [12]. The operators Di,j will play an important role for approximation theory on the sphere. We state several more properties of these operators. 20 FENG DAI AND YUAN XU

Lemma 8.3. For 1 i < j d, the operators D commute with ∆ . In particu- ≤ ≤ i,j 0 lar, D maps d to itself. i,j Hn Proof. By symmetry, we only to show only that D1,2. Let [A, B] = AB BA denote the commutator of A and B. A quick computation shows that − [∂ ,D ]= δ ∂ δ ∂ , [x ,D ]= δ x δ x , j k,l j,k l − j,l k i k,l i,k l − i,l k from which it is easy to see that, for 1 i < j d, 1 k

(8.7) f(x)Di,j g(x)dσ(x)= Di,j f(x)g(x)dσ(x). Sd−1 − Sd−1 Z Z Proof. By the rotation invariance of the Lebesgue measure dσ, we obtain, for every θ [ π, π], ∈ −

f(x)g(Qi,j,−θx) dσ(x)= f(Qi,j,θx)g(x) dσ(x). Sd−1 Sd−1 Z Z Differentiating both sides of this identity with respect to θ and evaluating the resulting equation at θ = 0 leads to, by (8.5), the desired equation (8.7). 

r Sd−1 The equation (8.7) allows us to define distributional derivatives Di,j on for r N via the identity, wtih g C∞(Sd−1), ∈ ∈ r r r (8.8) Di,j f(x)g(x)dσ(x) = ( 1) f(x)Di,j g(x)dσ(x). Sd−1 − Sd−1 Z Z Summing (8.8) with r = 2 over i < j and applying (8.3), it follows immediately that ∆0 is self-adjoint, which can also be deduced from Theorem 4.5. Corollary 8.5. For f,g C2(Sd−1), ∈

f(x)∆0g(x)dσ = ∆0f(x)g(x)dσ. Sd−1 Sd−1 Z Z The spherical gradient is a vector of first order differential operator on the ∇0 sphere, which can be written in terms of Di,j as follows. Lemma 8.6. For f C1(Sd−1) and 1 j d, the j-th component of f satisfies ∈ ≤ ≤ ∇0 (8.9) ( ) f(ξ)= ξ D f(ξ), ξ Sd−1. ∇0 j i i,j ∈ 1≤iX≤d,i6=j Furthermore, for f,g C1(Sd−1), the following identity holds: ∈ (8.10) f(ξ) g(ξ)= D f(ξ)D g(ξ), ξ Sd−1. ∇0 · ∇0 i,j i,j ∈ 1≤Xi

Proof. Let F (y)= f(y/ y ). From the definition and ξ = 1, we obtain k k k k ∂ d (8.11) ( ) f(ξ)= F (ξ)= ∂ f ξ ξ ∂ f ∇0 j ∂x j − j i i j i=1 X d d d = ∂ f ξ2 ξ ξ ∂ f = ξ D f, j i − j i i i i,j i=1 i=1 i=1 X X X which gives (8.9) since Di,if = 0. (8.11) means that (8.12) f(ξ)= f(ξ) ξ(ξ f). ∇0 ∇ − · ∇ Since ξ f(ξ) = 0, using (8.11) and the definition of D , it follows that · ∇0 i,j d d d f g = f g = ∂ f∂ g ξ ξ ∂ f∂ g = D f(ξ)D g(ξ), ∇0 · ∇0 ∇0 · ∇ j j − i j i j i,j i,j j=1 j=1 i=1 i

(8.13) f(x) 0g(x)dσ = ( 0f(x) (d 1)xf(x)) g(x)dσ. Sd−1 ∇ − Sd−1 ∇ − − Z Z Furthermore, for f C2(Sd−1) and g C1(Sd−1), ∈ ∈

(8.14) 0f 0gdσ = ∆0f(x)g(x)dσ. Sd−1 ∇ · ∇ − Sd−1 Z Z Proof. Using the expression (8.11) and applying (8.7), we obtain

f(x)( 0)j g(x)dσ = xif(x)Di,j g(x)dσ Sd−1 ∇ Sd−1 Z 1≤iX≤d,i6=j Z

= Di,j (xif(x))g(x)dσ. − Sd−1 1≤iX≤d,i6=j Z By the chain rule, recalling (8.2) if necessary, we have D (x f(x)) = x D f i,j i i i,j − xj f(x). Hence we obtain

f(x)( 0)j g(x)dσ = xiDi,j f(x) (d 1)f(x) g(x)dσ, Sd−1 ∇ − Sd−1 − − Z Z 1≤iX≤d,i6=j  which gives the j-th component of (8.13). Equation (8.14) follows immediately from (8.10), (8.7) and (8.3). 

9. Notes Spherical harmonics appear in many disciplines and in many different branches of mathematics. Many books contain parts of the theory of spherical harmonics. Our treatment covers what is needed for harmonic analysis and approximation theory in this book. Below we comment on some books that we have consulted. A classical treatise on spherical harmonics is [8], a good source for classical results. A short but nice expository is [9], which was later expanded into [10]. The reference book [5] contains a chapter on spherical harmonics. A rich resource for spherical harmonics in Fourier analysis is [11]. Applications to and connections 22 FENG DAI AND YUAN XU with group representation are studied extensively in [12]; see also [7]. For their role in the context of of several variables, see [4] as well as [5]. The book [1] contains a chapter on spherical harmonics in light of special functions. The theory of harmonic functions is treated in [3], including material on spherical harmonics. The book [6] deals with spherical harmonics in geometric applications. Finally, the recent book [2] provides an introduction to spherical harmonics and approximation on the sphere from the perspective of applications in numerical analysis. Aside from their role in representation theory, the operators Di,j do not seem to have received much attention in analysis. Most of the materials in Section 8 have not previously appeared in books. These operators play an important role in our development of approximation theory on the sphere. References

[1] R. Askey, G. Andrews and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. [2] K. Atikinson and W. Han, Spherical harmonics and approximation on the unit sphere: An introduction, Lecture Notes in Math. 2044, Springer, 2012. [3] S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory, Springer, New York, 1992. [4] C. F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge Univ. Press, 2001. [5] A. Erd´elyi, W. Magnus, F. Oberhettinger and Tricomi, F. G., Higher transcendental functions, McGraw-Hill, New York, 1953. [6] H. Groemer, Geometric applications of Fourier series and spherical harmonics, Cambridge University Press, New York, 1996. [7] S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. [8] E. W. Hobson, The theory of spherical and elliptical harmonics, Chelsea Publishing Co., New York, 1955. [9] C. M¨uller, Spherical harmonics. Lecture Notes in Math. No. 17, Springer-Verlag, New York, 1966. [10] C. M¨uller, Analysis of spherical symmetries in Euclidean spaces, Springer, New York, 1997 [11] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. [12] N. J. Vilenkin, Special functions and the theory of group representations, American Math- ematical Society Translation of Mathematics Monographs 22, American Mathematical Society, Providence, RI, 1968. 23 Approximation Theory and Harmonic Analysis on Spheres and Balls F. Dai and Y. Xu Springer Monographs in Mathematics, Springer, New York 2013.

Table of Contents

Preamble xvii

1 Spherical Harmonics 1 1.1 Spaceofsphericalharmonicsandorthogonalbases 1 1.2ProjectionoperatorsandZonalharmonics 7 1.3 Zonal basis of spherical harmonics 11 1.4 Laplace-Beltrami operator 14 1.5 Spherical harmonics in spherical coordinates 17 1.6 Spherical harmonics in two and three variables 19 1.6.1 Spherical harmonics in two variables 19 1.6.2 Spherical harmonics in three variables 20 1.7Representationoftherotationgroup 22 1.8 Angular derivatives and Laplace-Beltrami operator 23 1.9 Notes 27

2 Convolution and Spherical Harmonic Expansion 29 2.1 Convolution and translation operators 29 2.2 Fourier orthogonal expansions 33 2.3 The Hardy-Littlewood Maximal function 37 2.4SphericalharmonicseriesandCes`aromeans 40 2.5ConvergenceofCes`aromeans:furtherresults 43 2.6 highly localized kernels 44 2.7 Notes and further results 50

3 Littlewood-Paley Theory and Multiplier Theorem 53 3.1 Analysis on homogeneous spaces 53 3.2TheLittlewood-Paleytheoryonthesphere 56 3.3 The Marcinkiewitcz multiplier theorem 64 3.4 The Littlewood-Paley inequality 67 3.5 The Riesz transform on the sphere 71 3.5.1 Fractional Laplace-Beltrami operator and Riesz transform 71 3.5.2 Proof of Lemma 3.5.2 74 3.6 Notes and further results 78

4 Approximation on the Sphere 79 4.1 Approximation by trigonometric polynomials 80 4.2 Modulus of smoothness on the unit sphere 85 4.3 A key lemma 89 4.4 Characterization of best approximation 93 4.5 K-functionals and Approximation in Sobolev space 95 4.6 Computational examples 97 24 FENG DAI AND YUAN XU

4.7 Other moduli of smoothness 99 4.8 Notes and further results 101

5 Weighted Polynomial Inequalities 105 5.1 Doubling weights on the sphere 105 5.2 A maximal function for spherical polynomials 111 5.3 Marcinkiewicz-Zygmund inequalities 114 5.4 Further inequalities between sums and integrals 118 5.5 Nikolskii and Bernstein inequalities 124 5.6 Notes and further results 126

6 Cubature Formulas on Spheres 127 6.1 Cubature formulas 127 6.2 Product type cubature formulas on the sphere 132 6.3 Positive Cubature formulas 135 6.4 Area-regular partitions of Sd−1 140 6.5 Spherical designs 144 6.6 Notes and further results 151

7 Harmonic Analysis Associated to Reflection Groups 155 7.1 Dunkl operators and h-sphericalharmonics 156 7.2 Projection operator and intertwining operator 160 7.3 h-harmonicsforageneralfinitereflectiongroup 166 7.4 Convolution and h-harmonic series 168 7.5 Maximal function and multiplier theorem 172 Zd 7.6 Maximal function for 2 invariant weight 179 7.7 Notes and further results 186

8 BoundednessofProjectionOperatorandCes`aroMeans 189 8.1 BoundednessofCes`aromeansabovethecriticalindex 189 8.2 A multiple Beta integral of the 191 8.3 Pointwise estimate of the kernel functions 198 8.4 Proof of the main results 200 8.5 LowerboundforgeneralizedGegenbauerexpansion 204 8.6 Notes and further results 211

9 Projection Operators and Ces`aro Means in Lp Spaces 213 9.1Boundednessofprojectionoperators 213 9.2 Boundedness of Ces`aro means in Lp spaces 217 9.2.1 Proof of Theorem 9.2.1 218 9.2.2 Proof of Theorem 9.2.2 222 9.3Localestimatesoftheprojectionoperators 225 d−2 9.3.1 Proof of Theorem 9.1.2, case I: γ < σκ 2 226 − d−2 9.3.2 Proof of Theorem 9.1.2, case 2: γ = σκ 2 234 9.4 Notes and further results − 239

10 Weighted Best Approximation by Polynomials 241 10.1Moduliofsmoothnessandbestapproximation 241 10.2 Fractional powers of spherical h-Laplacian 245 10.3 K-functionals and best approximatioin 248 25

10.4 Equivalence of the first modulus with K-functional 250 10.5 Equivalence of the second modulus with K-functional 254 10.6Furtherpropertiesofmoduliofsmoothness 261 10.7 Notes and further results 262

11 Harmonic Analysis on the Unit Ball 265 11.1 Orthogonal structure on the unit ball 265 11.2Convolutionandorthogonalexpansions 272 11.3 Maximal functions and multiplier theorem 276 11.4ProjectionoperatorsandCe`saromeansontheball 280 11.5 Near best approximation operators and highly localized kernels 283 11.6 Cubature formulas on the unit ball 286 11.6.1 Cubature formulas on the ball and on the sphere 287 11.6.2 Positive cubature formulas and MZ inequality 288 11.6.3 Producttypecubatureformulas 290 11.7Orthogonalstructureonspheresandonballs 292 11.8 Notes and further results 294

12 Polynomial Approximation on the Unit Ball 297 12.1 Algebraic polynomial approximation on the interval 298 12.2 First modulus of smoothness and K-functional 303 12.2.1 Projection from sphere to ball 303 12.2.2 Modulus of smoothness 305 12.2.3 Weighted K-functional and the equivalence 307 12.2.4 Main theorems 309 12.2.5 The moduli of smoothness on [ 1, 1] 311 12.2.6 Computational examples − 312 12.3 The second modulus of smoothness and K-functional 313 12.3.1 Analogue of the Ditzian-Totik K-functional 314 12.3.2 Direct and inverse theorems by K-functional 319 12.3.3 Analogue of Ditzian-Totik modulus of smoothness on Bd 320 r 12.3.4 Equivalence of ωϕ(f,t)p and Kr,ϕ(f,t)p 323 12.4 Third modulus of smoothness and K-functional 326 12.5Comparisonsofthreemoduliofsmoothness 328 12.6 Notes and further results 329

13 Harmonic Analysis on the Simplex 333 13.1Orthogonalstructureonthesimplex 333 13.2Convolutionandorthogonalexpansions 337 13.3 Maximal functions and multiplier theorem 340 13.4ProjectionoperatorandCes`aromeans 344 13.5 Near best approximation operators and highly localized kernels 351 13.6 Weighted best approximation on the simplex 354 13.7 Cubature formulas on the simplex 356 13.7.1 Cubature formulas on the simplex and on the ball 357 13.7.2 Positive cubature formulas and MZ inequality 357 13.7.3 Producttypecubatureformulas 359 13.8 Notes and further results 360

14 Applications 361 26 FENG DAI AND YUAN XU

14.1 Highly localized tight polynomial frames on the sphere 362 14.2 Nodes distribution of positive cubature formulas 369 14.3 Positive definite functions on the sphere 373 14.4 Asymptotics for minimal discrete energy on the sphere 381 14.5 Computerized tomography 388 14.6 Notes and further results 396 ADistance,DifferenceandIntegralFormulas 401 A.1 Distance on spheres, balls and simplexes 401 A.2 Euler angles and rotations 403 A.3Basicpropertiesofdifferenceoperators 404 A.4Ces`aromeansandDifferenceoperators 406 A.5 Integrals over spheres and balls 409 BJacobiandRelatedOrthogonalPolynomials 413 B.1 Jacobi polynomials 413 B.2 Gegenbauer polynomials 416 B.3 Generalized Gegenbauer polynomials 418 B.4 Associated Legendre polynomials 419 B.5 Estimates of normalized Jacobi polynomials 419 References 425 Index 433 Symbol Index 437

Department of Mathematical and Statistical Sciences, University of Alberta, , Ed- monton, Alberta T6G 2G1, Canada. E-mail address: [email protected]

Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222. E-mail address: [email protected]