Spherical harmonics

Remarks

Example: on $S^2$, $Y(x, y, z) = x^2 - y^2$ is a spherical harmonic of degree $2$, since $\Delta(x^2 - y^2) = 2 - 2 = 0$. Non-example: $x^2$ is homogeneous of degree $2$ but $\Delta x^2 = 2 \ne 0$. On $S^2$ it equals $\left( x^2 - \frac13 (x^2 + y^2 + z^2) \right) + \frac13$, a sum of harmonics of degrees $2$ and $0$, so it is not a spherical harmonic of degree $2$. For $d \ge 3$, the dimension of $\mathcal{H}_\ell(S^{d-1})$ is $\binom{\ell + d - 1}{d - 1} - \binom{\ell + d - 3}{d - 1}$, which is $2\ell + 1$ for $d = 3$; for $d = 2$ it is $2$ when $\ell \ge 1$. Each $Y \in \mathcal{H}_\ell$ satisfies $\Delta_{S^{d-1}} Y = -\ell(\ell + d - 2) Y$ for the Laplace–Beltrami operator of the sphere. The spaces $\mathcal{H}_\ell$ are pairwise orthogonal in $L^2(S^{d-1})$, and their sum is dense. For $d = 3$, in spherical coordinates, a standard orthonormal basis is $Y_\ell^m(\theta, \varphi) = \sqrt{\frac{2\ell + 1}{4\pi} \frac{(\ell - m)!}{(\ell + m)!}} \, P_\ell^m(\cos \theta) \, e^{im\varphi}$ for $-\ell \le m \le \ell$, where $P_\ell^m$ are the associated Legendre polynomials.