Expansion in spherical harmonics

Remarks

Proof idea: on a smaller closed ball, $u$ is the Poisson integral of its boundary values. The Poisson kernel of the ball expands as a series of zonal harmonics, each a homogeneous harmonic polynomial in $x$; integrating term by term gives the $p_k$. Uniqueness follows by comparing homogeneous parts of the Taylor series of $u$ at $0$, since $p_k$ is the degree $k$ part. In polar form $x = r\zeta$ with $\abs{\zeta} = 1$, the expansion reads $u(r\zeta) = \sum_k r^k p_k(\zeta)$, and the restriction of $p_k$ to the unit sphere is a spherical harmonic of degree $k$. The restrictions to the sphere of harmonic polynomials of different degrees are orthogonal in $L^2$ of the sphere. In fact, $L^2$ of the sphere is the orthogonal direct sum of the spaces of spherical harmonics of each degree. Example: on the unit disc in $\R^2 = \C$, $u = \operatorname{Re} \frac{1}{1 - z}$ is harmonic, and $u(r e^{i\theta}) = \sum_{k \ge 0} r^k \cos k\theta$, where $r^k \cos k\theta = \operatorname{Re} z^k$ is a homogeneous harmonic polynomial of degree $k$.

Going deeper

For a student

This is the higher-dimensional version of the Fourier series of a harmonic function on the disc. It solves the Dirichlet problem on a ball by separation of variables: expand the boundary data in spherical harmonics and multiply the degree $k$ part by $(\abs{x}/R)^k$. In $\R^3$ the space of spherical harmonics of degree $k$ has dimension $2k + 1$, and the expansion is the one used for gravitational and electrostatic potentials. The Fischer decomposition describes how all polynomials, not only harmonic ones, split into harmonic pieces.