III.87 Spherical Harmonics

The starting point for FOURIER ANALYSIS [III.27] is the observation that a wide class of periodic functions f(θ) with period 2π can be decomposed as infinite linear combinations of the TRIGONOMETRIC FUNCTIONS [III.92] sinnθ and cosnθ, or, equivalently, as sums of the form a_ne^inθ.

A useful way to think of a periodic function f defined on the real line is as an equivalent function F defined on , the unit circle in the complex plane. A typical point on the circle has the form e^iθ, and we define F(e^iθ) to be f(θ). (Note that if we add 2π to θ then F(e^iθ) does not change because e^iθ = e^i(θ+2π) and f(θ) does not change because f is periodic with period 2π.)

If f(θ) = a_ne^inθ and we write z for e^iθ, then F(z) = a_nzⁿ. Therefore, if we consider functions defined on rather than periodic functions defined on , then Fourier analysis decomposes our functions into infinite linear combinations of the functions zⁿ, where n can be any integer.

What is special about the functions zⁿ? The answer is that they are the characters of , which means that they are the only nonzero continuous complex-valued functions defined on that satisfy the relation (zw) = (z)(w) for every z and w in .

Now imagine that F is a function defined not on but on the two-dimensional set S², which is the unit sphere in ³ (defined as the set of points (x,y,z) such that x² + y² + z² = 1). More generally, how about functions F defined on S^d-1 (defined as the set of points (x_l, . . . , x_d) such that )? Is there a natural way of decomposing such an F, at least if it is sufficiently nice? That is, is there a good way of generalizing Fourier analysis to higher-dimensional spheres?

There is an important and initially discouraging difference between the sphere S² and the circle S¹ = . We defined as a set of complex numbers rather than as a set of points in the plane ² because that way it forms a multiplicative group. The sphere, by contrast, does not have a useful group structure (for a clue about why, see QUATERNIONS, OCTONIONS, AND NORMED DIVISION ALGEBRAS [III.76]), so we cannot talk about characters. This makes it less obvious what the “nice” functions should be, into which we might hope to decompose more general functions.

However, there is another way of explaining why the trigonometric functions arise naturally, one that does not involve complex numbers. We can write a typical point in S¹ as (x,y) with x² + y² = 1, or equivalently as (cosθ, sinθ) for some real number θ. Then our basic functions, if we wish to avoid complex numbers, are cosnθ and sinnθ, but these can also be written in terms of x and y. For instance, cosθ and sinθ are x and y, respectively, cos2θ = cos²θ-sin²θ = x² - y², and so on. (Note that x² - y² = 2x² - 1 = 1 - 2y², since x² + y² = 1.) In general, cosnθ and sin nθ can always be written as polynomials in cos θ and sin θ, so the basic trigonometric functions can be thought of as restrictions to the unit circle of certain polynomials.

What are these polynomials? It turns out that they are harmonic and homogeneous. A harmonic polynomial p(x, y) is one that satisfies the LAPLACE EQUATION [I.3 §5.4] Δp = 0, where Δp stands for

For instance, if p(x, y) = x² - y², then ∂²p/∂x² = 2 and ∂²p/∂y² = -2, so x² - y² is, as we would hope, a harmonic polynomial. Since the Laplacian Δ is a linear operator, the harmonic polynomials form a vector space. A homogeneous polynomial of degree n is one in which the total degree of each term is n, or equivalently a polynomial p(x,y) such that p(λx,λy) is always equal to λⁿp(x, y). For example, x³ - 3xy² is homogeneous of degree 3 (and also harmonic). The homogeneous harmonic polynomials of degree n form a subspace of the space of all harmonic polynomials. It has dimension 1 when n = 0 and 2 when n > 0. (When n > 0 it corresponds to the space of functions of the form Acos nθ + Bsin nθ. The polynomial x³ - 3xy², for instance, corresponds to the function cos 3θ.)

The notion of a harmonic polynomial generalizes very easily to higher dimensions. For example, in three dimensions a harmonic polynomial is a polynomial p(x, y, z) such that

A spherical harmonic of order n and dimension d is the restriction to the sphere S^d-1 of a harmonic polynomial in d variables that is homogeneous of degree n.

Here are some of the properties of spherical harmonics that make them particularly useful and closely analogous to the trigonometric polynomials on the circle. We shall fix a dimension d and use the notation dμ to denote Haar measure on the unit sphere S = S^d-1. Basically, this means that if f is an integrable function from S to , then ∫_s f(x) dμ is its average.

(i) Orthogonality. If p and q are spherical harmonics of dimension d, and if they have different degrees, then ∫_s p(x)q(x) dμ = 0.

(ii) Completeness. Every function f : S → that belongs to L² (S, μ) (meaning that ∫_s |f(x)|²dμ exists and is finite) can be written as a sum H_n (with convergence in L²(S, μ)), where H_n is a spherical harmonic of order n.

(iii) Fihite-dimensionality of decomposition. For each d and n, the vector space of spherical harmonics of dimension d and order n is finite dimensional.

From these three properties it is easy to deduce that L² (S, μ) has an ORTHONORMAL BASIS [III.37] consisting of spherical harmonics.

Why are spherical harmonics natural, and why are they useful? Both questions can be given several answers: here is one for each.

The Laplace operator Δ, which operates on functions defined on ⁿ, can be generalized to functions defined on any RIEMANNIAN MANIFOLD [I.3 §6.10] M. The generalization, denoted Δ_M, is called the Laplace-Beltrami operator for M, and its behavior gives one a great deal of information about the geometry of M. In particular, the Laplace-Beltrami operator can be defined for the sphere S^d-1, where it is called simply the Beltrami operator. It turns out that the spherical harmonics are the EIGENVECTDRS [I.3 §4.3] of the Beltrami operator. More precisely, a spherical harmonic of dimension d and order n is an eigenvector with eigenvalue -n(n + d - 2). (Notice that the second derivative of cos nθ is -n² cos nθ, which corresponds to the case d = 2.) This gives an alternative, more natural (but less elementary) definition of spherical harmonics. This definition, combined with the fact that the Laplace operator is self-adjoint, explains many of the important properties of spherical harmonics. (See LINEAR OPERATORS AND THEIR PROPERTIES [III.50 §3] for an amplification of this remark.)

One reason for the importance of Fourier analysis is that many important linear operators become diagonal, and hence particularly easy to understand, when they are applied to the Fourier transform of a function. For example, if f is a smooth periodic function and we write it as Σ_n∈a_ne^inθ, then the derivative of f is Σ_n∈na_ne^inθ. Writing (n) and (n) for the nth Fourier coefficients of f and f′, respectively, we deduce that (n) = n(n), which tells us that to differentiate a function f all we have to do is multiply its Fourier transform pointwise by the function g(n) = n. This provides a very useful technique for solving differential equations.

As has already been mentioned, spherical harmonics are eigenvectors of the Laplacian, but they also diagonalize several other linear operators. A good example is the spherical Radon transform, which is defined as follows. If f is a function from S^d-1 to , then its spherical Radon transform Rf is another function from S^d-1 to , and the value of Rf at a point x is the average value of f over all points y that are orthogonal to x. This is closely related to the more usual Radon transform, which replaces a function defined on the plane by its averages over lines; inverting the Radon transform is important for creating images from the outputs of medical scanners. The spherical harmonics turn out to be eigenfunctions for the spherical Radon transform. More generally, any transform T of the form Tf(x) = ∫_S W(x · y)f(y)dμ(y), where w is a suitable function (or generalized function), is diagonalized by spherical harmonics. The eigenvalue associated with a given spherical harmonic can be calculated by the so-called Funk-Hecke formula.

Spherical harmonics give a way of linking CHEBYSHEV AND LEGENDRE POLYNOMIALS [III.85], and showing that both of them are natural concepts. The Chebyshev polynomials are those polynomials in x that are also spherical harmonics of dimension 2: that is, that are equal on S¹ to homogeneous harmonic polynomials in two variables. For instance, because x² + y² = 1 for every (x, y) in the circle S¹, the function x³ - 3xy² that we considered earlier is equal on S¹ to the function 4x³ - 3x, so 4x³ - 3x is a Chebyshev polynomial. The Legendre polynomials are those polynomialsin x that are equal to spherical harmonics of dimension 3. For example, if p (x,y, z) = 2x² - y² - z² then Δp = 0, and p(x,y, z) = 3x² - 1 everywhere on S², since x² + y² + z² = 1. Therefore, 3x² - 1 is a Legendre polynomial.

Here is a sketch of a proof that these polynomials are equal to the Chebyshev and Legendre polynomials as they are usually defined. The usual definition is that they are sequences of polynomials, one for each degree, that are uniquely determined by certain orthogonality relations. Because spherical harmonics of different orders are orthogonal, the polynomials just described also satisfy certain orthogonality relations. When one works out what these are, one discovers that they are precisely the relations that define the Chebyshev and Legendre polynomials.