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Notes on Spherical Harmonics and Linear Representations of Lie Groups

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Notes on Spherical Harmonics and Linear Representations of Lie Groups Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] November 13, 2013 2 Contents 1 Spherical Harmonics 5 1.1 Introduction, Spherical Harmonics on the Circle . 5 1.2 Spherical Harmonics on the 2-Sphere . 8 1.3 The Laplace-Beltrami Operator . 15 1.4 Harmonic Polynomials, Spherical Harmonics and L2(Sn) . 22 1.5 Spherical Functions and Representations of Lie Groups . 31 1.6 Reproducing Kernel and Zonal Spherical Functions . 38 1.7 More on the Gegenbauer Polynomials . 47 1.8 The Funk-Hecke Formula . 50 1.9 Convolution on G=K, for a Gelfand Pair (G; K) . 53 3 4 CONTENTS Chapter 1 Spherical Harmonics and Linear Representations of Lie Groups 1.1 Introduction, Spherical Harmonics on the Circle In this chapter, we discuss spherical harmonics and take a glimpse at the linear representa- tion of Lie groups. Spherical harmonics on the sphere, S2, have interesting applications in computer graphics and computer vision so this material is not only important for theoretical reasons but also for practical reasons. Joseph Fourier (1768-1830) invented Fourier series in order to solve the heat equation [12]. Using Fourier series, every square-integrable periodic function, f, (of period 2π) can be expressed uniquely as the sum of a power series of the form 1 X f(θ) = a0 + (ak cos kθ + bk cos kθ); k=1 where the Fourier coefficients, ak; bk, of f are given by the formulae 1 Z π 1 Z π 1 Z π a0 = f(θ) dθ; ak = f(θ) cos kθ dθ; bk = f(θ) sin kθ dθ; 2π −π π −π π −π for k ≥ 1. The reader will find the above formulae in Fourier's famous book [12] in Chapter III, Section 233, page 256, essentially using the notation that we use nowdays. This remarkable discovery has many theoretical and practical applications in physics, signal processing, engineering, etc. We can describe Fourier series in a more conceptual manner if we introduce the following inner product on square-integrable functions: Z π hf; gi = f(θ)g(θ) dθ; −π 5 6 CHAPTER 1. SPHERICAL HARMONICS which we will also denote by Z hf; gi = f(θ)g(θ) dθ; S1 where S1 denotes the unit circle. After all, periodic functions of (period 2π) can be viewed as functions on the circle. With this inner product, the space L2(S1) is a complete normed 1 vector space, that is, a Hilbert space. Furthermore, if we define the subspaces, Hk(S ), 2 1 1 1 of L (S ), so that H0(S ) (= R) is the set of constant functions and Hk(S ) is the two- dimensional space spanned by the functions cos kθ and sin kθ, then it turns out that we have a Hilbert sum decomposition 1 2 1 M 1 L (S ) = Hk(S ) k=0 S1 1 2 1 into pairwise orthogonal subspaces, where k=0 Hk(S ) is dense in L (S ). The functions 1 cos kθ and sin kθ are also orthogonal in Hk(S ). 1 Now, it turns out that the spaces, Hk(S ), arise naturally when we look for homoge- neous solutions of the Laplace equation, ∆f = 0, in R2 (Pierre-Simon Laplace, 1749-1827). Roughly speaking, a homogeneous function in R2 is a function that can be expressed in polar coordinates, (r; θ), as f(r; θ) = rkg(θ): Recall that the Laplacian on R2 expressed in cartesian coordinates, (x; y), is given by @2f @2f ∆f = + ; @x2 @y2 where f : R2 ! R is a function which is at least of class C2. In polar coordinates, (r; θ), where (x; y) = (r cos θ; r sin θ) and r > 0, the Laplacian is given by 1 @ @f 1 @2f ∆f = r + : r @r @r r2 @θ2 If we restrict f to the unit circle, S1, then the Laplacian on S1 is given by @2f ∆ 1 f = : s @θ2 1 2 It turns out that the space Hk(S ) is the eigenspace of ∆S1 for the eigenvalue −k . To show this, we consider another question, namely, what are the harmonic functions on R2, that is, the functions, f, that are solutions of the Laplace equation, ∆f = 0: Our ancestors had the idea that the above equation can be solved by separation of variables. This means that we write f(r; θ) = F (r)g(θ) , where F (r) and g(θ) are independent functions. 1.1. INTRODUCTION, SPHERICAL HARMONICS ON THE CIRCLE 7 To make things easier, let us assume that F (r) = rk, for some integer k ≥ 0, which means that we assume that f is a homogeneous function of degree k. Recall that a function, f : R2 ! R, is homogeneous of degree k iff f(tx; ty) = tkf(x; y) for all t > 0: Now, using the Laplacian in polar coordinates, we get 1 @ @(rkg(θ)) 1 @2(rkg(θ)) ∆f = r + r @r @r r2 @θ2 1 @ @2g = krkg + rk−2 r @r @θ2 @2g = rk−2k2g + rk−2 @θ2 k−2 2 = r (k g + ∆S1 g): Thus, we deduce that 2 ∆f = 0 iff ∆S1 g = −k g; 2 that is, g is an eigenfunction of ∆S1 for the eigenvalue −k . But, the above equation is equivalent to the second-order differential equation d2g + k2g = 0; dθ2 whose general solution is given by g(θ) = an cos kθ + bn sin kθ: 2 In summary, we found that the integers, 0; −1; −4; −9;:::; −k ;::: are eigenvalues of ∆S1 and that the functions cos kθ and sin kθ are eigenfunctions for the eigenvalue −k2, with k ≥ 0. So, it looks like the dimension of the eigenspace corresponding to the eigenvalue −k2 is 1 when k = 0 and 2 when k ≥ 1. It can indeed be shown that ∆S1 has no other eigenvalues and that the dimensions claimed for the eigenspaces are correct. Observe that if we go back to our homogeneous harmonic functions, f(r; θ) = rkg(θ), we see that this space is spanned by the functions k k uk = r cos kθ; vk = r sin kθ: Now, (x + iy)k = rk(cos kθ + i sin kθ), and since <(x + iy)k and =(x + iy)k are homogeneous polynomials, we see that uk and vk are homogeneous polynomials called harmonic polyno- mials. For example, here is a list of a basis for the harmonic polynomials (in two variables) of degree k = 0; 1; 2; 3; 4: k = 0 1 k = 1 x; y k = 2 x2 − y2; xy k = 3 x3 − 3xy2; 3x2y − y3 k = 4 x4 − 6x2y2 + y4; x3y − xy3: 8 CHAPTER 1. SPHERICAL HARMONICS Therefore, the eigenfunctions of the Laplacian on S1 are the restrictions of the harmonic 2 1 2 1 L1 1 polynomials on R to S and we have a Hilbert sum decomposition, L (S ) = k=0 Hk(S ). It turns out that this phenomenon generalizes to the sphere Sn ⊆ Rn+1 for all n ≥ 1. Let us take a look at next case, n = 2. 1.2 Spherical Harmonics on the 2-Sphere The material of section is very classical and can be found in many places, for example Andrews, Askey and Roy [1] (Chapter 9), Sansone [25] (Chapter III), Hochstadt [17] (Chapter 6) and Lebedev [21] (Chapter ). We recommend the exposition in Lebedev [21] because we find it particularly clear and uncluttered. We have also borrowed heavily from some lecture notes by Hermann Gluck for a course he offered in 1997-1998. Our goal is to find the homogeneous solutions of the Laplace equation, ∆f = 0, in R3, 2 and to show that they correspond to spaces, Hk(S ), of eigenfunctions of the Laplacian, ∆S2 , on the 2-sphere, 2 3 2 2 2 S = f(x; y; z) 2 R j x + y + z = 1g: 2 Then, the spaces Hk(S ) will give us a Hilbert sum decomposition of the Hilbert space, L2(S2), of square-integrable functions on S2. This is the generalization of Fourier series to 2 the 2-sphere and the functions in the spaces Hk(S ) are called spherical harmonics. The Laplacian in R3 is of course given by @2f @2f @2f ∆f = + + : @x2 @y2 @z2 If we use spherical coordinates x = r sin θ cos ' y = r sin θ sin ' z = r cos θ; in R3, where 0 ≤ θ < π, 0 ≤ ' < 2π and r > 0 (recall that ' is the so-called azimuthal angle in the xy-plane originating at the x-axis and θ is the so-called polar angle from the z-axis, angle defined in the plane obtained by rotating the xz-plane around the z-axis by the angle '), then the Laplacian in spherical coordinates is given by 1 @ 2 @f 1 ∆f = r + ∆ 2 f; r2 @r @r r2 S where 1 @ @f 1 @2f ∆ 2 f = sin θ + ; S sin θ @θ @θ sin2 θ @'2 1.2. SPHERICAL HARMONICS ON THE 2-SPHERE 9 is the Laplacian on the sphere, S2 (for example, see Lebedev [21], Chapter 8 or Section 1.3, where we derive this formula). Let us look for homogeneous harmonic functions, f(r; θ; ') = rkg(θ; '), on R3, that is, solutions of the Laplace equation ∆f = 0: We get k 1 @ 2 @(r g) 1 k ∆f = r + ∆ 2 (r g) r2 @r @r r2 S 1 @ k+1 k−2 = kr g + r ∆ 2 g r2 @r S k−2 k−2 = r k(k + 1)g + r ∆S2 g k−2 = r (k(k + 1)g + ∆S2 g): Therefore, ∆f = 0 iff ∆S2 g = −k(k + 1)g; that is, g is an eigenfunction of ∆S2 for the eigenvalue −k(k + 1).
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